Regular Schur labeled skew shape posets and their 0-Hecke modules

Abstract

Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$. We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$. Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf {M}_P$ are also discussed.

1 Introduction

Schur labeled skew shape posets naturally appear in the context of the celebrated Stanley’s P-partitions conjecture. Let $\mathsf {P}_n$ be the set of posets with underlying set $[n] := \{1,2,\ldots , n\}$ . Each poset $P \in \mathsf {P}_n$ can be identified with the labeled poset $(P, \omega )$ with the labeling $\omega : P \rightarrow [n]$ given by $\omega (i) = i$ . Consequently, to each poset $P\in \mathsf {P}_n$ , one can associate the following generating function for its P-partitions:

$$\begin{align*}K_{P} := \sum_{f: P\text{-partition}} x_1^{|f^{-1}(1)|} x_2^{|f^{-1}(2)|} \cdots. \end{align*}$$

In 1972, Stanley [34, p. 81] proposed a conjecture stating that $K_P$ is a symmetric function if and only if P is a Schur labeled skew shape poset. For the precise definition of Schur labeled skew shape posets, refer to Section 2.3. While this conjecture has been verified to be true for all posets P with $|P| \le 8$ , it remains an open question in the general case (see [28]). We denote by $\mathsf {SP}_n$ the set of all Schur labeled skew shape posets in $\mathsf {P}_n$ .

Regular posets were introduced by Björner–Wachs [8] during their investigation of the convex subsets of the symmetric group $\mathfrak {S}_n$ on $\{1,2, \ldots n\}$ under the right weak Bruhat order. For $P \in \mathsf {P}_n$ with the partial order $\preceq $ , let $\Sigma _R(P)$ be the set of permutations $\pi \in \mathfrak {S}_n$ satisfying that if $x \preceq y$ , then $\pi ^{-1}(x) \le \pi ^{-1}(y)$ . They observed that every convex subset of $\mathfrak {S}_n$ under the right weak Bruhat order appears as $\Sigma _R(P)$ for some $P \in \mathsf {P}_n$ , and every right weak Bruhat interval in $\mathfrak {S}_n$ is convex. This observation led them to characterize the posets $P \in \mathsf {P}_n$ satisfying that $\Sigma _R(P)$ is a right weak Bruhat interval. They introduced the notion of regular posets and proved that $P \in \mathsf {P}_n$ is a regular poset if and only if $\Sigma _R(P)$ is a right weak Bruhat interval in $\mathfrak {S}_n$ . For the definition of regular posets, refer to Definition 2.3. We denote by $\mathsf {RP}_n$ the set of all regular posets in $\mathsf {P}_n$ .

Let $\mathsf {RSP}_n := \mathsf {RP}_n \cap \mathsf {SP}_n$ . In the following, we explain the reason why we consider regular Schur labeled skew shape posets from the perspective of the representation theory of the $0$ -Hecke algebra.

In 1996, Duchamp, Krob, Leclerc and Thibon [12] showed that the Grothendieck ring of the tower of $0$ -Hecke algebras $\bigoplus _{n\ge 0}H_n(0)$ , when equipped with addition and multiplication from direct sum and induction product, is isomorphic to the ring $\mathrm {QSym}$ of quasisymmetric functions. To be precise, they showed that the map

$$ \begin{align*} \mathrm{ch} : \bigoplus_{n \ge 0} \mathcal{G}_0(H_n(0)\text{-}\mathbf{mod}) \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_\alpha] \mapsto F_{\alpha}, \end{align*} $$

called the quasisymmetric characteristic, is a ring isomorphism. Here, $\mathcal {G}_0(H_n(0)\text{-}\mathbf {mod})$ is the Grothendieck group of the category $H_n(0)\text{-}\mathbf {mod}$ of finitely generated left $H_n(0)$ -modules, $\alpha $ is a composition, $\mathbf {F}_\alpha $ is the irreducible $H_n(0)$ -module attached to $\alpha $ , and $F_\alpha $ is the fundamental quasisymmetric function attached to $\alpha $ (for more details, see Section 2.4). Afterwards, Bergeron–Li [5] showed that the map $\mathrm {ch}$ is not just a ring isomorphism but also a Hopf algebra isomorphism. In 2002, Duchamp–Hivert–Thibon [11] associated a right $H_n(0)$ -module $M_P$ with each poset $P \in \mathsf {P}_n$ , such that the image of $M_P$ under the quasisymmetric characteristic is $K_P$ . This was achieved by defining a suitable right $H_n(0)$ -action on $\Sigma _R(P)$ .

Since the middle of 2010, various left $0$ -Hecke modules, each equipped with a tableau basis and yielding an important quasisymmetric characteristic image, have been constructed ([2, 4, 32, 37, 38]). In order to handle these modules in a uniform manner, Jung–Kim–Lee–Oh [19] introduced a left $H_n(0)$ -module $\mathsf {B}(I)$ , referred to as the weak Bruhat interval module associated with I, for each left weak Bruhat interval I in $\mathfrak {S}_n$ . Furthermore, they showed that $\bigoplus _{n \ge 0} \mathcal {G}_0(\mathscr {B}_n)$ is isomorphic to $\mathrm {QSym}$ as Hopf algebras, where $\mathscr {B}_n$ is the full subcategory of $H_n(0)$ - $\mathbf {mod}$ consisting of objects that are direct sums of finitely many isomorphic copies of weak Bruhat interval modules of $H_n(0)$ . Recently, Choi–Kim–Oh [9] clarified the exact relationship between the weak Bruhat interval modules and the $0$ -Hecke modules $M_P$ , using Björner–Wachs’ characterization. More precisely, they constructed a contravariant functor $\mathcal {F}:{H_n(0)\text{-}\mathbf {mod}} \rightarrow \mathbf {mod}\text{-}H_n(0)$ that preserves the quasisymmetric characteristic and showed that $M_P = \mathcal {F}(\mathsf {B}(\Sigma _L(P)))$ , where $\mathbf {mod}\text{-}H_n(0)$ is the category of finitely generated right $H_n(0)$ -modules and $\Sigma _L(P) := \{\gamma ^{-1} \mid \gamma \in \Sigma _R(P)\}$ for $P \in \mathsf {RP}_n$ . For technical reasons, we use a slightly different $0$ -Hecke module, denoted as $\mathsf {M}_P$ , instead of Duchamp, Hivert and Thibon’s module $M_P$ . This module is a left $H_n(0)$ -module with the basis $\Sigma _L(P)$ . For the detailed definition of $\mathsf {M}_P$ , refer to Definition 2.8.

The aim of this paper is to give a comprehensive investigation of regular Schur labeled skew shape posets and their associated $0$ -Hecke modules.

In Section 3, we provide an explicit description of $\Sigma _L(P)$ for $P \in \mathsf {RSP}_n$ . We first introduce a Schur labeling $\tau _P$ , which is a bijective tableau uniquely determined by suitable conditions. For details, see Equation (3.2). Let $\lambda /\mu $ be the shape of $\tau _P$ . Then $\tau _P$ gives rise to a reading, denoted $\mathsf {read}_{\tau _P}$ , on the set $\mathrm {SYT}(\lambda /\mu )$ of standard Young tableaux of shape $\lambda /\mu $ . We show that all permutations in $\Sigma _L(P)$ appear as reading words of standard Young tableaux of shape $\lambda /\mu $ , i.e., $ \Sigma _L(P) = \mathsf {read}_{\tau _P}(\mathrm {SYT}(\lambda /\mu )) $ (Lemma 3.2). Then, we derive that

$$\begin{align*}\Sigma_L(P) = [\mathsf{read}_{\tau_P}(T_{\lambda/\mu}), \mathsf{read}_{\tau_P}(T^{\prime}_{\lambda/\mu})]_L,\end{align*}$$

where $T_{\lambda /\mu }$ (resp. $T^{\prime}_{\lambda /\mu }$ ) is the standard Young tableau obtained by filling the Young diagram of shape $\lambda /\mu $ by $1, 2, \ldots , n$ from left to right starting with the top row (resp. from top to bottom starting with leftmost column) (Theorem 3.9).

In Section 4, we introduce an equivalence relation $\overset {D}{\simeq }$ on the set $\mathrm {Int}(n)$ of left weak Bruhat intervals in $\mathfrak {S}_n$ . This relation is defined by $I_1 \overset {D}{\simeq } I_2$ if there is a descent-preserving poset isomorphism between $I_1$ and $I_2$ . We show that every equivalence class C is of the form

$$\begin{align*}\{[\gamma, \xi_C\gamma]_L \mid \gamma \in [\sigma_0, \sigma_1]_R\}, \end{align*}$$

where $\sigma _0$ and $\sigma _1$ are the minimal and maximal elements in $\{\sigma \mid [\sigma , \rho ]_L \in C\}$ , respectively, and $\xi _C = \rho \sigma ^{-1}$ for any $[\sigma , \rho ]_L \in C$ (Theorem 4.6). In the case where $P \in \mathsf {RSP}_n$ , we show in Theorem 4.7 that the equivalence class of $\Sigma _L(P)$ is given by

(1.1) $$ \begin{align} \{\Sigma_L(Q) \mid Q \in \mathsf{RSP}_n\text{ with }\mathrm{sh}(\tau_Q) = \mathrm{sh}(\tau_P) \}. \end{align} $$

In Section 5, we classify the $H_n(0)$ -modules $\mathsf {M}_P$ up to isomorphism as P ranges over $\mathsf {RSP}_n$ . We show in Theorem 5.5 that for $P, Q \in \mathsf {RSP}_n$ ,

$$ \begin{align*} \mathsf{M}_P \cong \mathsf{M}_Q \quad \text{if and only if} \quad \mathrm{sh}(\tau_P) = \mathrm{sh}(\tau_Q). \end{align*} $$

The ‘if’ part is straightforward and can be derived from Equation (1.1). As for the ‘only if’ part, it can be verified by showing that when $\tau _P$ and $\tau _Q$ have different shapes, it results in either nonisomorphic projective covers or nonisomorphic injective hulls of $\mathsf {M}_P$ and $\mathsf {M}_Q$ . To accomplish this, we compute both a projective cover and an injective hull of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ (Lemma 5.4).

In Section 6, we first prove that a poset $P \in \mathsf {P}_n$ is a regular Schur labeled skew shape poset if and only if $\Sigma _L(P)$ is dual plactic-closed (Theorem 6.4). This improves Malvenuto’s result [27, Theorem 1], which states that if $\Sigma _L(P)$ is dual plactic-closed, then $P \in \mathsf {SP}_n$ . Then, we introduce the notion of a distinguished filtration of an $H_n(0)$ -module M with respect to a linearly independent subset $\mathcal {B}$ of $\mathrm {QSym}_n$ (Definition 6.5). If such a filtration is available, we have a representation theoretic interpretation of the expansion of $\mathrm {ch}([M])$ in $\mathcal {B}$ . The existence of a distinguished filtration is quite nontrivial as seen in Example 6.6. However, using the characterization given in Theorem 6.4, we show that $\mathsf {M}_P$ admits a distinguished filtration with respect to the Schur basis when $P \in \mathsf {RSP}_n$ (Theorem 6.7).

The final section is mainly devoted to further issues concerned with the classification and decomposition of the $0$ -Hecke modules $\mathsf {M}_P$ . We discuss the classification problem for $\{\mathsf {M}_P \mid P \in \mathsf {SP}_n\}$ and $\{\mathsf {M}_P \mid P \in \mathsf {RP}_n\}$ . In particular, we expect that for $P, Q \in \mathsf {RP}_n$ , $\mathsf {M}_P \cong \mathsf {M}_Q$ if and only if $\Sigma _L(P) \overset {D}{\simeq } \Sigma _L(Q)$ (Conjecture 7.2). The decomposition problem is also discussed for the $0$ -Hecke modules $\mathsf {M}_P$ when $P \in \mathsf {RSP}_n$ . Based on experimental data, we expect that for $P \in \mathsf {RSP}_n$ , $\mathsf {M}_P$ is indecomposable if and only if $\mathrm {sh}(\tau _P)$ is disconnected and does not contain any disconnected ribbon (Conjecture 7.5). At the end of this section, we provide a remark on how to recover $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ from a module of the generic Hecke algebra $H_n(q)$ by specializing q to $0$ .

In the appendix, we give a tableau description of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ . For a skew partition $\lambda /\mu $ of size n, we construct an $H_n(0)$ -module $X_{\lambda /\mu }$ with standard Young tableaux of shape $\lambda /\mu $ as basis elements. This module can be viewed as a representative of the isomorphism class of $\mathsf {M}_P$ in the category $H_n(0)\text{-}\mathbf {mod}$ for every $P \in \mathsf {RSP}_n$ with $\mathrm {sh}(\tau _P) = \lambda /\mu $ .

2 Preliminaries

For integers m and n, we define $[m,n]$ and $[n]$ to be the intervals $\{t\in \mathbb Z \mid m\le t \le n\}$ and $\{t\in \mathbb Z \mid 1\le t \le n\}$ , respectively. Throughout this paper, n will denote a nonnegative integer unless otherwise stated.

2.1 Compositions, Young diagrams and bijective tableaux

A composition $\alpha $ of n, denoted by $\alpha \models n$ , is a finite ordered list of positive integers $(\alpha _1,\alpha _2,\ldots , \alpha _k)$ satisfying $\sum _{i=1}^k \alpha _i = n$ . We call $\alpha _i$ ( $1 \le i \le k$ ) a part of $\alpha $ , $k =: \ell (\alpha )$ the length of $\alpha $ , and $n =:|\alpha |$ the size of $\alpha $ . And we define the empty composition $\varnothing $ to be the unique composition of size and length $0$ . Whenever necessary, we set $\alpha _i = 0$ for all $i> \ell (\alpha )$ .

Given $\alpha = (\alpha _1,\alpha _2,\ldots ,\alpha _k) \models n$ and $I = \{i_1 < i_2 < \cdots < i_{l}\} \subseteq [n-1]$ , let

$$ \begin{align*} \mathrm{set}(\alpha) &:= \{\alpha_1,\alpha_1+\alpha_2,\ldots, \alpha_1 + \alpha_2 + \cdots + \alpha_{k-1}\}, \text{ and} \\\mathrm{comp}(I) &:= (i_1,i_2 - i_1, i_3 - i_2, \ldots,n-i_{l}). \end{align*} $$

The set of compositions of n is in bijection with the set of subsets of $[n-1]$ under the correspondence $\alpha \mapsto \mathrm {set}(\alpha )$ (or $I \mapsto \mathrm {comp}(I)$ ). The reverse composition $\alpha ^{\mathrm {r}}$ of $\alpha $ is defined to be the composition $(\alpha _k, \alpha _{k-1}, \ldots , \alpha _1)$ , and the complement $\alpha ^{\mathrm {c}}$ of $\alpha $ is defined to be the unique composition satisfying $\mathrm {set}(\alpha ^c) = [n-1] \setminus \mathrm {set}(\alpha )$ .

If a composition $\lambda = (\lambda _1, \lambda _2, \ldots , \lambda _k) \models n$ satisfies $\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _k$ , then it is called a partition of n and denoted as $\lambda \vdash n$ . Given two partitions $\lambda $ and $\mu $ with $\ell (\lambda ) \ge \ell (\mu )$ , we write $\lambda \supseteq \mu $ if $\lambda _i \geq \mu _i$ for all $1 \le i \le \ell (\mu )$ . A skew partition $\lambda /\mu $ is a pair $(\lambda , \mu )$ of partitions with $\lambda \supseteq \mu $ . We call $|\lambda /\mu | := |\lambda | - |\mu |$ the size of $\lambda /\mu $ . In the case where $\lambda \supset \mu \supset \nu $ , we say that $\lambda /\mu $ extends $\mu /\nu $ .

Given a partition $\lambda $ , we define the Young diagram $\mathtt {yd}(\lambda )$ of $\lambda $ to be the left-justified array of n boxes, where the ith row from the top has $\lambda _i$ boxes for $1 \le i \le k$ . Similarly, given a skew partition $\lambda / \mu $ , we define the Young diagram $\mathtt {yd}(\lambda / \mu )$ of $\lambda / \mu $ to be the Young diagram $\mathtt {yd}(\lambda )$ with all boxes belonging to $\mathtt {yd}(\mu )$ removed. A Young diagram is called connected if for each pair of consecutive rows, there are at least two boxes (one in each row) which have a common edge. A skew partition is called connected if the corresponding Young diagram is connected, and it is called basic if the corresponding Young diagram contains neither empty rows nor empty columns. In this paper, every skew partition is assumed to be basic unless otherwise stated.

For two skew partitions $\lambda /\mu $ and $\nu /\kappa $ , we define $\lambda /\mu \star \nu /\kappa $ to be the skew partition whose Young diagram is obtained by taking a rectangle of empty squares with the same number of rows as $\mathtt {yd}(\lambda /\mu )$ and the same number of columns as $\mathtt {yd}(\nu /\kappa )$ , and putting $\mathtt {yd}(\nu /\kappa )$ below and $\mathtt {yd}(\lambda /\mu )$ to the right of this rectangle. For instance, if $\lambda /\mu = (2,2)$ and $\nu /\kappa = (3,2)/(1)$ , then $\lambda /\mu \star \nu /\kappa = (5,5,3,2)/(3,3,1)$ and

Given a skew partition $\lambda /\mu $ of size n, a bijective tableau of shape $\lambda / \mu $ is a filling of $\mathtt {yd}(\lambda / \mu )$ with distinct entries in $[n]$ . For later use, we denote by $\tau _0^{\lambda /\mu }$ (resp. $\tau _1^{\lambda /\mu }$ ) the bijective tableau of shape $\lambda / \mu $ obtained by filling $1, 2, \ldots , n$ from right to left starting with the top row (resp. from top to bottom starting with the rightmost column). If $\lambda /\mu $ is clear in the context, we will drop the superscript $\lambda /\mu $ from $\tau _0^{\lambda /\mu }$ and $\tau _1^{\lambda /\mu }$ . Again, letting $\lambda / \mu = (2,2) \star (3,2) / (1)$ , we have

A bijective tableau is referred to as a standard Young tableau if the elements in each row are arranged in increasing order from left to right, and the elements in each column are arranged in increasing order from top to bottom. We denote by $\mathrm {SYT}(\lambda / \mu )$ the set of all standard Young tableaux of shape $\lambda / \mu $ . And we let $\mathrm {SYT}_n := \bigcup _{\lambda \vdash n} \mathrm {SYT}(\lambda )$ .

2.2 Weak Bruhat orders on the symmetric group

Let $\mathfrak {S}_n$ denote the symmetric group on $[n]$ . Every permutation $\sigma \in \mathfrak {S}_n$ can be expressed as a product of simple transpositions $s_i := (i,i+1)$ for $1 \leq i \leq n-1$ . A reduced expression for $\sigma $ is an expression that represents $\sigma $ in the shortest possible length, and the length $\ell (\sigma )$ of $\sigma $ is the number of simple transpositions in any reduced expression for $\sigma $ . Let

$$\begin{align*}\mathrm{Des}_L(\sigma):= \{i \in [n-1] \mid \ell(s_i \sigma) < \ell(\sigma)\} \ \ \text{and} \ \ \mathrm{Des}_R(\sigma):= \{i \in [n-1] \mid \ell(\sigma s_i) < \ell(\sigma)\}. \end{align*}$$

It is well known that if $\sigma = w_1 w_2 \cdots w_n$ in one-line notation, then

$$ \begin{align*} \begin{aligned} \mathrm{Des}_L(\sigma) & = \{ i \in [n-1] \mid i\text{ is right of }i+1\text{ in }w_1 w_2 \cdots w_n \} \quad \text{and} \\ \mathrm{Des}_R(\sigma) & = \{ i \in [n-1] \mid w_i> w_{i+1} \}. \end{aligned} \end{align*} $$

The left weak Bruhat order $\preceq _L$ (resp. right weak Bruhat order $\preceq _R$ ) on $\mathfrak {S}_n$ is the partial order on $\mathfrak {S}_n$ whose covering relation $\preceq _L^{\mathrm {c}}$ (resp. $\preceq _R^{\mathrm {c}}$ ) is given as follows:

$$\begin{align*}\sigma \preceq_L^{\mathrm{c}} s_i \sigma \text{ if }i \notin \mathrm{Des}_L(\sigma) \quad \text{(resp. }\sigma \preceq_R^{\mathrm{c}} \sigma s_i\text{ if }i \notin \mathrm{Des}_R(\sigma)). \end{align*}$$

Although these two weak Bruhat orders are not identical, there exists a poset isomorphism

$$ \begin{align*}(\mathfrak{S}_n, \preceq_L)\to (\mathfrak{S}_n, \preceq_R),\quad \sigma \mapsto \sigma^{-1}.\end{align*} $$

For each $\gamma \in \mathfrak {S}_n$ , let

$$ \begin{align*} \mathrm{Inv}_L(\gamma) & := \{(i,j) \mid 1 \le i < j \le n \text{ and } \gamma(i)> \gamma(j) \} \quad \text{and} \\ \mathrm{Inv}_R(\gamma) & := \{(\gamma(i), \gamma(j)) \mid 1 \le i < j \le n \text{ and } \gamma(i)> \gamma(j)\}. \end{align*} $$

Then, for $\sigma , \rho \in \mathfrak {S}_n$ ,

$$ \begin{align*} & \sigma \preceq_L \rho \quad \text{if and only if} \quad \mathrm{Inv}_L(\sigma) \subseteq \mathrm{Inv}_L(\rho) \quad \text{and}\\ & \sigma \preceq_R \rho \quad \text{if and only if} \quad \mathrm{Inv}_R(\sigma) \subseteq \mathrm{Inv}_R(\rho). \end{align*} $$

Given $\sigma , \rho \in \mathfrak {S}_n$ , the left weak Bruhat interval $[\sigma , \rho ]_L$ (resp. the right weak Bruhat interval $[\sigma , \rho ]_R$ ) denotes the closed interval $\{\gamma \in \mathfrak {S}_n \mid \sigma \preceq _L \gamma \preceq _L \rho \}$ (resp. $\{\gamma \in \mathfrak {S}_n \mid \sigma \preceq _R \gamma \preceq _R \rho \}$ ) with respect to the left weak Bruhat order (resp. the right weak Bruhat order).

For later use, we introduce the following lemma.

Lemma 2.1 [6, Proposition 3.1.6]

For $\sigma ,\rho \in \mathfrak {S}_n$ with $\sigma \preceq _R \rho $ , the map $[\sigma , \rho ]_R \rightarrow [\mathrm {id}, \sigma ^{-1}\rho ]_R, \gamma \mapsto \sigma ^{-1} \gamma $ is a poset isomorphism. Equivalently, for $\sigma ,\rho \in \mathfrak {S}_n$ with $\sigma \preceq _L \rho $ , the map $[\sigma ,\rho ]_L \rightarrow [\mathrm {id}, \rho \sigma ^{-1}]_L,\gamma \mapsto \gamma \sigma ^{-1}$ is a poset isomorphism.

Let us collect notations which will be used later. For $S \subseteq \mathfrak {S}_n$ and $\xi \in \mathfrak {S}_n$ , let

$$\begin{align*}S \cdot \xi := \{\gamma \xi \mid \gamma \in S \} \quad \text{and} \quad \xi \cdot S := \{ \xi \gamma \mid \gamma \in S \}. \end{align*}$$

We use $w_0$ to denote the longest element in $\mathfrak {S}_n$ . For $I \subseteq [n-1]$ , let $\mathfrak {S}_{I}$ be the parabolic subgroup of $\mathfrak {S}_n$ generated by $\{s_i \mid i\in I\}$ and $w_0(I)$ the longest element in $\mathfrak {S}_{I}$ . For $\alpha \models n$ , let $w_0(\alpha ) := w_0(\mathrm {set}(\alpha ))$ . Finally, for $\sigma \in \mathfrak {S}_n$ , we let $\sigma ^{w_0} := w_0 \sigma w_0$ .

Lemma 2.2 [7, Theorem 6.2]

For $I \subseteq J \subseteq [n-1]$ , we have

$$\begin{align*}\{\sigma \in \mathfrak{S}_n \mid I \subseteq \mathrm{Des}_L(\sigma) \subseteq J\} = [w_0(I), w_0 (J^{\mathrm{c}}) w_0]_R. \end{align*}$$

2.3 Regular posets and Schur labeled skew shape posets

Let $\mathsf {P}_n$ be the set of posets whose underlying set is $[n]$ . Given $P \in \mathsf {P}_n$ , we write the partial order of P as $\preceq _P$ .

Definition 2.3 [8, p. 110]

A poset $P \in \mathsf {P}_n$ is said to be regular if the following holds: for all $x,y,z \in [n]$ with $x \preceq _P z$ , if $x < y < z$ or $z<y<x$ , then $x \preceq _P y$ or $y \preceq _P z$ .

We denote by $\mathsf {RP}_n$ the set of all regular posets in $\mathsf {P}_n$ . In the following, we will explain how regular posets can be characterized in terms of left weak Bruhat intervals.

Given $P \in \mathsf {P}_n$ , let

$$\begin{align*}\Sigma_{L}(P) := \{\sigma \in \mathfrak{S}_n \mid \sigma(i) \leq \sigma(j)\text{ for all }i, j\in [n]\text{ with }i \preceq_P j\}. \end{align*}$$

Throughout this paper, $\Sigma _L(P)$ is considered as the set of all linear extensions of P under the correspondence $\sigma \mapsto ([n], \preceq _E)$ , where $\preceq _E$ is the total order on $[n]$ given by $\sigma ^{-1}(1) \preceq _E \sigma ^{-1}(2) \preceq _E \cdots \preceq _E \sigma ^{-1}(n)$ .

Theorem 2.4 [8, Theorem 6.8]

Let $U \subseteq \mathfrak {S}_n$ with $|U|> 1$ . The following conditions are equivalent:

1. (1) U is a left weak Bruhat interval.

2. (2) $U = \Sigma _L(P)$ for some $P \in \mathsf {RP}_n$ .

Consider the map

$$\begin{align*}\unicode{x3b7}: \mathsf{P}_n \rightarrow \mathscr{P}(\mathfrak{S}_n), \quad P \mapsto \Sigma_L(P), \end{align*}$$

where $\mathscr {P}(\mathfrak {S}_n)$ is the power set of $\mathfrak {S}_n$ . One can see that $\unicode{x3b7} $ is injective. Combining this with Theorem 2.4, we obtain a one-to-one correspondence

$$\begin{align*}\unicode{x3b7}|_{\mathsf{RP}_n}: \mathsf{RP}_n \rightarrow \mathrm{Int}(n), \quad P \mapsto \Sigma_L(P), \end{align*}$$

where $\mathrm {Int}(n)$ is the set of nonempty left weak Bruhat intervals in $\mathfrak {S}_n$ .

Next, let us introduce Schur labeled skew shape posets. Let $\lambda / \mu $ be a skew partition of size n. Given a bijective tableau $\tau $ of shape $\lambda /\mu $ , we define $\mathsf {poset}(\tau )$ to be the poset $([n], \preceq _\tau )$ , where

(2.1) $$ \begin{align} i \preceq_\tau j\text{ if and only if }i\text{ lies weakly upper-left of }j\text{ in }\tau. \end{align} $$

The Hasse diagram of $\mathsf {poset}(\tau )$ can be obtained by rotating $\tau 135^\circ $ counterclockwise.¹

Example 2.5. Let $\lambda / \mu = (2,2) \star (3,2) / (1)$ . For the bijective tableaux $\tau _0$ and $\tau _1$ of shape $\lambda /\mu $ introduced in Section 2.1, we have

A Schur labeling of shape $\lambda / \mu $ is a bijective tableau of shape $\lambda / \mu $ such that the entries in each row decrease from left to right and the entries in each column increase from top to bottom. Let $\mathsf {S}(\lambda /\mu )$ be the set of all Schur labelings of shape $\lambda / \mu $ . Since $\tau _0$ and $\tau _1$ are Schur labelings of shape $\lambda /\mu $ , $\mathsf {S}(\lambda /\mu )$ is nonempty. Set

$$\begin{align*}\mathsf{SP}(\lambda/\mu) := \{\mathsf{poset}(\tau) \mid \tau \in \mathsf{S}(\lambda/\mu) \} \quad \text{and} \quad \mathsf{SP}_n := \bigcup_{ |\lambda/\mu| = n } \mathsf{SP}(\lambda/\mu). \end{align*}$$

Definition 2.6. A poset $P \in \mathsf {P}_n$ is said to be a Schur labeled skew shape poset if it is contained in $\mathsf {SP}_n$ .

Remark 2.7. In some papers, for instance [27, 28], authors used a different convention than ours for Schur labeling. We adopt the definition of Schur labeling used in Stanley’s paper [34].

For simplicity, we set $\mathsf {RSP}_n := \mathsf {RP}_n \cap \mathsf {SP}_n$ .

2.4 The $0$ -Hecke algebra and the quasisymmetric characteristic

The $0$ -Hecke algebra $H_n(0)$ is the associative $\mathbb C$ -algebra with $1$ generated by $\pi _1,\pi _2,\ldots ,\pi _{n-1}$ subject to the following relations:

$$ \begin{align*} \pi_i^2 &= \pi_i \quad \text{for }1\le i \le n-1,\\ \pi_i \pi_{i+1} \pi_i &= \pi_{i+1} \pi_i \pi_{i+1} \quad \text{for }1\le i \le n-2,\\ \pi_i \pi_j &=\pi_j \pi_i \quad \text{if }|i-j| \ge 2. \end{align*} $$

For each $1 \leq i \leq n-1$ , let $\overline {\pi }_i := \pi _i - 1$ . Then, $\{\overline {\pi }_i \mid i = 1, 2, \ldots , n-1\}$ is also a generating set of $H_n(0)$ .

For any reduced expression $s_{i_1} s_{i_2} \cdots s_{i_p}$ for $\sigma \in \mathfrak {S}_n$ , let

$$\begin{align*}\pi_{\sigma} := \pi_{i_1} \pi_{i_2 } \cdots \pi_{i_p} \quad \text{and} \quad \overline{\pi}_{\sigma} := \overline{\pi}_{i_1} \overline{\pi}_{i_2} \cdots \overline{\pi}_{i_p}. \end{align*}$$

It is well known that these elements are independent of the choices of reduced expressions, and both $\{\pi _\sigma \mid \sigma \in \mathfrak {S}_n\}$ and $\{\overline {\pi }_\sigma \mid \sigma \in \mathfrak {S}_n\}$ are $\mathbb C$ -bases for $H_n(0)$ .

According to [30], there are $2^{n-1}$ pairwise nonisomorphic irreducible $H_n(0)$ -modules which are naturally indexed by compositions of n. To be precise, for each composition $\alpha $ of n, there exists an irreducible $H_n(0)$ -module $\mathbf {F}_{\alpha }:=\mathbb C v_{\alpha }$ endowed with the $H_n(0)$ -action defined as follows: for each $1 \le i \le n-1$ ,

$$\begin{align*}\pi_i \cdot v_\alpha = \begin{cases} 0 & i \in \mathrm{set}(\alpha),\\ v_\alpha & i \notin \mathrm{set}(\alpha). \end{cases} \end{align*}$$

Let $H_n(0)\text{-}\mathbf {mod}$ be the category of finite dimensional left $H_n(0)$ -modules and $\mathcal {R}(H_n(0))$ the $\mathbb Z$ -span of the set of (representatives of) isomorphism classes of modules in $H_n(0)\text{-}\mathbf {mod}$ . We denote by $[M]$ the isomorphism class corresponding to an $H_n(0)$ -module M. The Grothendieck group $\mathcal {G}_0(H_n(0))$ of $H_n(0)$ - $\mathbf {mod}$ is the quotient of $\mathcal {R}(H_n(0))$ modulo the relations $[M] = [M'] + [M"]$ whenever there exists a short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M" \rightarrow 0$ . The equivalence classes of the irreducible $H_n(0)$ -modules form a $\mathbb Z$ -basis for $\mathcal {G}_0(H_n(0))$ . Let

$$\begin{align*}\mathcal{G} := \bigoplus_{n \ge 0} \mathcal{G}_0(H_n(0)). \end{align*}$$

Let us review the connection between $\mathcal {G}$ and the ring $\mathrm {QSym}$ of quasisymmetric functions. For the definition of quasisymmetric functions, see [35, Section 7.19]. For a composition $\alpha $ , the fundamental quasisymmetric function $F_\alpha $ , which was firstly introduced in [16], is defined by

$$\begin{align*}F_\varnothing = 1 \quad \text{and} \quad F_\alpha = \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_n \\ i_j < i_{j+1} \text{ if } j \in \mathrm{set}(\alpha)}} x_{i_1}x_{i_2} \cdots x_{i_n} \quad \text{if }\alpha \neq \varnothing. \end{align*}$$

It is known that $\{F_\alpha \mid \alpha \text { is a composition}\}$ is a $\mathbb Z$ -basis for $\mathrm {QSym}$ . When M is an $H_m(0)$ -module and N is an $H_n(0)$ -module, we write $M \boxtimes N$ for the induction product of M and N; that is,

$$ \begin{align*} M \boxtimes N := M \otimes N \uparrow_{H_m(0) \otimes H_n(0)}^{H_{m+n}(0)}. \end{align*} $$

Here, $H_m(0) \otimes H_n(0)$ is viewed as the subalgebra of $H_{m+n}(0)$ generated by $\{\pi _i \mid i \in [m+n-1] \setminus \{m\} \}$ . The induction product induces a multiplication on $\mathcal {G}$ . It was shown in [12] that the linear map

$$ \begin{align*} \mathrm{ch} : \mathcal{G} \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_{\alpha}] \mapsto F_{\alpha}, \end{align*} $$

called quasisymmetric characteristic, is a ring isomorphism. Indeed, it turns out to be a Hopf algebra isomorphism when $\mathcal {G}$ has the comultiplication induced from restriction.

It is well known that $H_n(0)$ has the automorphisms $\unicode{x3b8} $ and $\unicode{x3c6} $ , as well as the anti-automorphism $\unicode{x3c7} $ , defined in the following manner:

$$ \begin{align*} &\unicode{x3c6}: H_n(0) \rightarrow H_n(0), \quad \pi_i \mapsto \pi_{n-i} \quad \text{for }1 \le i \le n-1,\\ &\unicode{x3b8}: H_n(0) \rightarrow H_n(0), \quad \pi_i \mapsto - \overline{\pi}_i \quad \text{for }1 \le i \le n-1, \\ &\unicode{x3c7}: H_n(0) \rightarrow H_n(0), \quad \pi_i \mapsto \pi_i \quad \text{for }1 \le i \le n-1 \end{align*} $$

(for instance, see [13, 22, 24, 25]). These maps commute with each other.

Note that an automorphism $\mu $ of $H_n(0)$ induces a covariant functor

$$\begin{align*}\mathbf{T}^{+}_\mu: H_n(0)\text{-}\mathbf{mod} \rightarrow H_n(0)\text{-}\mathbf{mod} \end{align*}$$

called the $\mu $ -twist. Similarly, an anti-automorphism $\nu $ of $H_n(0)$ induces a contravariant functor

$$\begin{align*}\mathbf{T}^{-}_\nu: H_n(0)\text{-}\mathbf{mod} \rightarrow H_n(0)\text{-}\mathbf{mod} \end{align*}$$

called the $\nu $ -twist. For the precise definitions of $\mathbf {T}^{+}_\mu $ and $\mathbf {T}^{-}_\nu $ , see [19, Subsection 3.4]. In [13, Proposition 3.3.], it was shown that

$$\begin{align*}\mathbf{T}^{+}_\unicode{x3c6}(\mathbf{F}_\alpha) = \mathbf{F}_{\alpha^{\mathrm{r}}}, \quad \mathbf{T}^{+}_\unicode{x3b8}(\mathbf{F}_\alpha) = \mathbf{F}_{\alpha^{\mathrm{c}}}, \quad \text{and} \quad \mathbf{T}^{-}_\unicode{x3c7}(\mathbf{F}_\alpha) = \mathbf{F}_{\alpha} \end{align*}$$

for $\alpha \models n$ . Let $\unicode{x3c1} $ and $\unicode{x3c8} $ be the automorphisms of $\mathrm {QSym}$ defined by

$$ \begin{align*} \unicode{x3c1}(F_\alpha) = F_{\alpha^{\mathrm{r}}} \quad \text{and} \quad \unicode{x3c8}(F_\alpha) = F_{\alpha^{\mathrm{c}}} \end{align*} $$

for every composition $\alpha $ . For a finite dimensional $H_n(0)$ -module M, it holds that

(2.2) $$ \begin{align} \begin{array}{c} \mathrm{ch}( [\mathbf{T}^{+}_\unicode{x3c6}(M)] ) = \unicode{x3c1} \circ \mathrm{ch}([M]), \quad \mathrm{ch} ( [\mathbf{T}^{+}_\unicode{x3b8}(M)] ) = \unicode{x3c8} \circ \mathrm{ch}([M]),\\[1ex] \text{and} \quad \mathrm{ch} ( [\mathbf{T}^{-}_\unicode{x3c7}(M)] ) = \mathrm{ch}([M]). \end{array} \end{align} $$

2.5 Modules arising from posets and weak Bruhat interval modules

Let $P\in \mathsf {P}_n$ . In [11, Definition 3.18], Duchamp, Hivert and Thibon defined a right $H_n(0)$ -module $M_P$ associated with P. In this paper, we are primarily concerned with left modules, so we introduce a left $H_n(0)$ -module, denoted as $\mathsf {M}_P$ , associated with P.

Definition 2.8. Let $P \in \mathsf {P}_n$ . Define $\mathsf {M}_P$ to be the left $H_n(0)$ -module with $\mathbb C \Sigma _{L}(P)$ as the underlying space and with the $H_n(0)$ -action defined by

(2.3) $$ \begin{align} \pi_{i} \cdot \gamma := \begin{cases} \gamma & \text{if }i \in \mathrm{Des}_L(\gamma), \\ 0 & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \notin \Sigma_{L}(P), \\ s_i \gamma & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \in \Sigma_{L}(P). \end{cases} \end{align} $$

One can see that the $H_n(0)$ -action provided in Equation (2.3) is well-defined through a slight modification of the proof in [11, Subsection 3.9] that $M_P$ is a well-defined right $H_n(0)$ -module. Indeed, there is a close connection between $\mathsf {M}_P$ and $M_P$ . Let $\mathbf {mod}\text{-}H_n(0)$ be the category of finite dimensional right $H_n(0)$ -modules. In [9, Subsection 4.3], the authors introduced a contravariant functor

$$ \begin{align*}\mathcal{F}_n: H_n(0)\text{-}\mathbf{mod} \rightarrow \mathbf{mod}\text{-}H_n(0)\end{align*} $$

that preserves quasisymmetric characteristics.² Using this functor, it is not difficult to see that

$$\begin{align*}\mathsf{M}_P \cong \mathbf{T}^{+}_\unicode{x3b8} \circ \mathbf{T}^{-}_\unicode{x3c7} \circ \mathcal{F}_n^{-1}(M_P). \end{align*}$$

Since the underlying set of P is $[n]$ , we can regard P as the labeled poset $(P, \omega )$ with the labeling $\omega : P \rightarrow [n]$ given by $\omega (i) = i$ . Under this consideration, a map $f: [n] \rightarrow \mathbb Z_{\ge 0}$ is called a P-partition if it satisfies the following conditions:

1. (1) If $i \preceq _P j$ , then $f(i) \le f(j)$ .

2. (2) If $i \preceq _P j$ and $i> j$ , then $f(i) < f(j)$ .

We define the P-partition generating function $K_P$ of P by

$$\begin{align*}K_{P} := \sum_{f: P\text{-partition}} x_1^{|f^{-1}(1)|} x_2^{|f^{-1}(2)|} \cdots. \end{align*}$$

Theorem 2.9. For $P \in \mathsf {P}_n$ , the following hold.

1. (1) $\mathrm {ch}([\mathsf {M}_P]) = \unicode{x3c8} (K_{P})$ .

2. (2) If $P \in \mathsf {SP}(\lambda /\mu )$ for a skew partition $\lambda /\mu $ , then $\mathrm {ch}([\mathsf {M}_P]) = s_{\lambda /\mu }$ .

Proof. (1) It was shown in [11, Theorem 3.21(i)] that the (right) quasisymmetric characteristic of $M_P$ is given by $K_P$ . Since $\mathsf {M}_P \cong \mathbf {T}^{+}_\unicode{x3b8} \circ \mathbf {T}^{-}_\unicode{x3c7} \circ \mathcal {F}_n^{-1}(M_P)$ , the assertion follows from Equation (2.2).

(2) In the same manner as in [35, Subsection 7.19], one can see that $K_P = s_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}}$ . Here, $\lambda ^{\mathrm {t}}$ and $\mu ^{\mathrm {t}}$ are the transposes of $\lambda $ and $\mu $ , respectively. Now the assertion can be derived from the well-known identity $\unicode{x3c8} (s_{\lambda /\mu }) = s_{\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}}}$ (for instance, see [26, Subsection 3.6]).

Next, let us introduce weak Bruhat interval modules, which were introduced by Jung, Kim, Lee and Oh [19] to provide a unified method for dealing with $H_n(0)$ -modules constructed using tableaux.

Definition 2.10 [19, Definition 1]

For each left weak Bruhat interval $[\sigma , \rho ]_L$ in $\mathfrak {S}_n$ , define $\mathsf {B}([\sigma , \rho ]_L)$ (simply, $\mathsf {B}(\sigma , \rho )$ ) to be the $H_n(0)$ -module with $\mathbb C[\sigma ,\rho ]_L$ as the underlying space and with the $H_n(0)$ -action defined by

$$ \begin{align*} \pi_i \cdot \gamma := \begin{cases} \gamma & \text{if }i \in \mathrm{Des}_L(\gamma), \\ 0 & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \notin [\sigma,\rho]_L, \\ s_i \gamma & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \in [\sigma,\rho]_L. \end{cases} \end{align*} $$

This module is called the weak Bruhat interval module associated to $[\sigma ,\rho ]_L$ .

We can deduce from Theorem 2.4 that for every $[\sigma , \rho ]_L \in \mathrm {Int}(n)$ , there exists a unique poset $P\in \mathsf {P}_n$ such that $\Sigma _L(P)=[\sigma , \rho ]_L$ . Since both $\mathsf {B}(\sigma , \rho )$ and $\mathsf {M}_P$ share $[\sigma , \rho ]_L$ as their basis and exhibit identical $H_n(0)$ -actions on this set, we can conclude that $\mathsf {B}(\sigma , \rho )$ is indeed equal to $\mathsf {M}_P$ .

Remark 2.11. The weak Bruhat interval modules are equipped with the structure of semi-combinatorial $H_n(0)$ -modules due to Hivert, Novelli and Thibon [17] and also that of diagram modules due to Searles [33]. More precisely,

1. (1) $\mathsf {B}(\sigma ,\rho )$ is the semi-combinatorial $H_n(0)$ -module associated to the Yang-Baxter interval $[Y_{\sigma }(\mathrm {id}), Y_{\rho }(\mathrm {id})]$ , and

2. (2) it was shown in [33, Subsection 7.3] that $\mathsf {B}(\sigma ,\rho )$ is isomorphic to a diagram module $\widehat {\mathbf {N}}_{\mathsf {StdTab}(D)}$ .

3 The weak Bruhat interval structure of $\Sigma _L(P)$ for $P \in \mathsf {RSP}_n$

Let $P \in \mathsf {RSP}_n$ . In this section, we explicitly describe the left weak Bruhat interval $\Sigma _L(P)$ in terms of reading words of standard Young tableaux. To begin with, we introduce readings for bijective tableaux.

Definition 3.1. Let $\tau $ be a bijective tableau of shape $\lambda / \mu $ . The $\tau $ -reading is the map

$$ \begin{align*}\mathsf{read}_\tau: \{\text{bijective tableaux of shape }\lambda / \mu\} \rightarrow \mathfrak{S}_n, \quad T \mapsto \mathsf{read}_\tau(T), \end{align*} $$

where $\mathsf {read}_\tau (T)$ is the permutation in $\mathfrak {S}_n$ given by $\mathsf {read}_\tau (T)(k) = T_{\tau ^{-1}(k)}$ for $1 \le k \le n$ . We call $\mathsf {read}_\tau (T)$ the $\tau $ -reading word of T.

Given a bijective tableaux T of shape $\lambda / \mu $ , the permutation $\mathsf {read}_{\tau }(T)$ in one-line notation can be obtained by reading the entries of T in the order given by $\tau ^{-1}(1), \tau ^{-1}(2), \ldots , \tau ^{-1}(n)$ . For instance, if and , then $\mathsf {read}_{\tau }(T) = 53412$ . With this definition, we have the following lemma.

Lemma 3.2. For any bijective tableau $\tau $ of shape $\lambda / \mu $ , $\Sigma _L(\mathsf {poset}(\tau )) = \mathsf {read}_\tau (\mathrm {SYT}(\lambda /\mu ))$ .

Proof. We first show that $\mathsf {read}_\tau (\mathrm {SYT}(\lambda /\mu )) \subseteq \Sigma _L(\mathsf {poset}(\tau ))$ . To do this, take any $T \in \mathrm {SYT}(\lambda /\mu )$ and $i,j \in [n]$ with $i \preceq _\tau j$ (for the definition of $\preceq _\tau $ , see Equation (2.1). Let $B_1$ and $B_2$ be the boxes in $\mathtt {yd}(\lambda /\mu )$ such that $\tau _{B_1} = i$ and $\tau _{B_2} = j$ . Since $i \preceq _\tau j$ , $B_1$ is weakly upper-left of $B_2$ in $\mathtt {yd}(\lambda /\mu )$ . This implies that

$$ \begin{align*}\mathsf{read}_\tau(T)(i) = T_{B_1} \leq T_{B_2} = \mathsf{read}_\tau(T)(j). \end{align*} $$

Therefore, $\mathsf {read}_\tau (T) \in \Sigma _L(\mathsf {poset}(\tau ))$ .

To complete the proof, let us show that $|\Sigma _L(\mathsf {poset}(\tau ))| = |\mathsf {read}_\tau (\mathrm {SYT}(\lambda /\mu ))|$ . Note that

$$ \begin{align*} \mathrm{ch}([\mathsf{M}_{\mathsf{poset}(\tau^{\lambda/\mu}_0)}]) = \sum_{\gamma \in \Sigma_L(\mathsf{poset}(\tau^{\lambda/\mu}_0))} F_{\mathrm{comp}(\mathrm{Des}_L(\gamma))^{\mathrm{c}}} \quad \text{and} \quad \mathrm{ch}([\mathsf{M}_{\mathsf{poset}(\tau^{\lambda/\mu}_0)}]) = s_{\lambda/\mu}, \end{align*} $$

where the second equality follows from Theorem 2.9(2). Putting these together with the well-known equality $s_{\lambda / \mu } = \sum _{T \in \mathrm {SYT}(\lambda / \mu )} F_{\mathrm {comp}(T)}$ , we have

(3.1) $$ \begin{align} \sum_{\gamma \in \Sigma_L(\mathsf{poset}(\tau^{\lambda/\mu}_0))} F_{\mathrm{comp}(\mathrm{Des}_L(\gamma))^{\mathrm{c}}} = \sum_{T \in \mathrm{SYT}(\lambda/\mu)} F_{\mathrm{comp}(T)}. \end{align} $$

Here, $\mathrm {comp}(T) = \mathrm {comp}(\{i \in [n-1] \mid i\text { is weakly right of }i+1\text { in }T\})$ . As a consequence of Equation (3.1), we have

$$\begin{align*}|\Sigma_L(\mathsf{poset}(\tau))| = |\Sigma_L(\mathsf{poset}(\tau^{\lambda/\mu}_0))| = |\mathrm{SYT}(\lambda/\mu)| = |\mathsf{read}_\tau(\mathrm{SYT}(\lambda/\mu))|.\\[-42pt] \end{align*}$$

The purpose of the remainder of this section is to describe the minimum and maximum of $\Sigma _L(P)$ with respect to $\preceq _L$ .

As a first step, we deal with a characterization of regular Schur labeled skew shape posets. For this purpose, the following definition is necessary.

Definition 3.3. Let $\lambda /\mu $ be a skew partition of size n. A Schur labeling $\tau $ of shape $\lambda / \mu $ is said to be distinguished if $\tau _B \geq \tau _{B'}$ whenever B is weakly below and weakly left of $B'$ for boxes $B, B' \in \mathtt {yd}(\lambda /\mu )$ .

Example 3.4. Consider the Schur labelings of shape $(3,2,2)/(1)$

One sees that $\tau _0^{(3,2,2)/(1)}$ and $\tau _1^{(3,2,2)/(1)}$ are distinguished, whereas $\tau $ is non-distinguished since $1$ appears weakly below and weakly left of $2$ in $\tau $ .

Let $\mathsf {DS}(\lambda / \mu )$ be the set of all distinguished Schur labelings of shape $\lambda / \mu $ . For any Schur labeling $\tau $ , let $\mathsf {cnt}_i(\tau )$ be the set of entries in the ith connected component of $\tau $ from the top. For each $P \in \mathsf {SP}_n$ , there exists a unique Schur labeling $\tau $ such that

(3.2) $$ \begin{align} &(\text{i})\ \mathrm{sh}(\tau)\text{ is basic},\nonumber\\ &(\text{ii})\ \mathsf{poset}(\tau) = P,\text{ and}\\ &(\text{iii})\ \mathrm{min}(\mathsf{cnt}_i(\tau)) < \mathrm{min}(\mathsf{cnt}_j(\tau))\text{ for }1 \leq i < j \leq k,\text{ where }k\text{ is the number of connected components}\nonumber\\ &\quad\text{of }P.\nonumber \end{align} $$

We denote this Schur labeling as $\tau _P$ . One can easily see that for $P \in \mathsf {SP}_n$ , $\tau _P$ is distinguished if and only if every connected component of $\tau _P$ is filled with consecutive integers.

Example 3.5. Given two Schur labeled skew shape posets

we have that

Lemma 3.6. For $P \in \mathsf {SP}_n$ , P is regular if and only if $\tau _P$ is distinguished.

Proof. To prove the ‘only if’ part, assume that P is a regular Schur labeled skew shape poset and $\lambda / \mu $ is the shape of $\tau _P$ . We claim that every connected component of $\tau _P$ is filled with consecutive integers. Take an arbitrary connected component $\mathsf {C}$ of $\tau _P$ . Let $B_1$ be the box at the top of the rightmost column of $\mathsf {C}$ and $B_2$ the box at the bottom of the leftmost column of $\mathsf {C}$ . Then, we may choose boxes $A_0:= B_1, A_2, A_3, \ldots , A_k := B_2$ satisfying that for all $1 \le i \le k-1$ , $A_{i+1}$ is weakly below and weakly left of $A_i$ and $A_{i}, A_{i+1}$ are in the same row or in the same column. Let $m \in [(\tau _P)_{B_1}, (\tau _P)_{B_2}]$ . Then, there exists a unique index $1 \leq i \leq k-1$ such that $(\tau _P)_{A_i} \leq m \leq (\tau _P)_{A_{i+1}}$ . Since P is regular, one of the following holds:

1. (i) If $(\tau _P)_{A_i} \preceq _P (\tau _P)_{A_{i+1}}$ , then $(\tau _P)_{A_i} \preceq _P m$ or $m \preceq _P (\tau _P)_{A_{i+1}}$ .

2. (ii) If $(\tau _P)_{A_{i+1}} \preceq _P (\tau _P)_{A_{i}}$ , then $(\tau _P)_{A_{i+1}} \preceq _P m$ or $m \preceq _P (\tau _P)_{A_{i}}$ .

It follows that $(\tau _P)_{A_i}$ , m and $(\tau _P)_{A_{i+1}}$ appear in the same connected component; that is, m appears in $\mathsf {C}$ . Thus, $\tau _P$ is distinguished.

Next, to prove the ‘if’ part, assume that $\tau _P$ is a distinguished Schur labeling and $\lambda / \mu $ is the shape of $\tau _P$ . Let $B_1, B_2 \in \mathtt {yd}(\lambda /\mu )$ with $(\tau _P)_{B_1} \prec _{\tau _P} (\tau _P)_{B_2}$ . By the definition of $\prec _{\tau _P}$ , $B_1$ and $B_2$ are in the same connected component. In order to establish the regularity of P, we need to prove that either $(\tau _P)_{B_1} \preceq _{\tau _P} (\tau _P)_C$ or $(\tau _P)_{C} \preceq _{\tau _P} (\tau _P)_{B_2}$ for all $C \in \mathtt {yd}(\lambda /\mu )$ satisfying $(\tau _P)_{B_1} < (\tau _P)_C < (\tau _P)_{B_2}$ or $(\tau _P)_{B_1}> (\tau _P)_C > (\tau _P)_{B_2}$ .

Assume that there exists $C \in \mathtt {yd}(\lambda /\mu )$ such that $(\tau _P)_{B_1} < (\tau _P)_C < (\tau _P)_{B_2}$ . Since $\tau _P$ is a Schur labeling and $(\tau _P)_{B_1} \prec _{\tau _P} (\tau _P)_{B_2}$ , the inequality $(\tau _P)_{B_1} < (\tau _P)_{B_2}$ implies that $B_2$ is strictly below $B_1$ . In addition, since $\tau _P$ is distinguished and $(\tau _P)_{B_1},(\tau _P)_{B_2}$ appear in the same connected component in $\tau _P$ , $(\tau _P)_{C}$ appears in the same connected component with them in $\tau _P$ . Suppose for the sake of contradiction that $(\tau _P)_{B_1} \not \preceq _{\tau _P} (\tau _P)_C$ and $(\tau _P)_{C} \not \preceq _{\tau _P} (\tau _P)_{B_2}$ . Then C satisfies one of the following conditions:

1. (i) C is strictly above $B_1$ and strictly right of $B_2$ .

2. (ii) C is strictly left of $B_1$ and strictly below $B_2$ .

However, since $\tau _P$ is a Schur labeling and $(\tau _P)_{B_1} < (\tau _P)_C$ , C cannot satisfy (i). Similarly, since $\tau _P$ is a Schur labeling and $(\tau _P)_{C} < (\tau _P)_{B_2}$ , C cannot satisfy (ii). Therefore, $(\tau _P)_{B_1} \preceq _{\tau _P} (\tau _P)_C$ or $(\tau _P)_{C} \preceq _{\tau _P} (\tau _P)_{B_2}$ . In a similar way, one can show that if there exists $C \in \mathtt {yd}(\lambda /\mu )$ such that $(\tau _P)_{B_1}> (\tau _P)_C > (\tau _P)_{B_2}$ , then $(\tau _P)_{B_1} \preceq _{\tau _P} (\tau _P)_C$ or $(\tau _P)_{C} \preceq _{\tau _P} (\tau _P)_{B_2}$ . Thus, P is regular.

Note that $\tau _{\mathsf {poset}(\tau )} = \tau $ for any $\tau \in \mathsf {DS}(\lambda / \mu )$ . Considering this property together with Lemma 3.6, one can see that the map

(3.3) $$ \begin{align} \Phi: \mathsf{RSP}_n \rightarrow \bigcup_{|\lambda/\mu| = n} \mathsf{DS}(\lambda/\mu), \quad P \mapsto \tau_P \end{align} $$

is a bijection and its inverse is given by $\tau \mapsto \mathsf {poset}(\tau )$ .

As a second step, we provide a lemma that will be used throughout this paper.

Lemma 3.7. For $T \in \mathrm {SYT}(\lambda /\mu )$ , $\{\mathsf {read}_\tau (T) \mid \tau \in \mathsf {DS}(\lambda /\mu )\} = [\mathsf {read}_{\tau _0}(T), \mathsf {read}_{\tau _1}(T)]_R$ .

Proof. Let us show the inclusion $\{\mathsf {read}_\tau (T) \mid \tau \in \mathsf {DS}(\lambda /\mu )\} \subseteq [\mathsf {read}_{\tau _0}(T), \mathsf {read}_{\tau _1}(T)]_R$ . This can be done by proving $\mathsf {read}_{\tau _0}(T) \preceq _R \mathsf {read}_{\tau }(T)$ and $\mathsf {read}_{\tau }(T) \preceq _R \mathsf {read}_{\tau _1}(T)$ for all $\tau \in \mathsf {DS}(\lambda /\mu )$ . Since the method of proof for the latter inequality is essentially the same as that for the former one, we omit the proof for the latter inequality. Let $\tau \in \mathsf {DS}(\lambda /\mu )$ and $(i,j) \in \mathrm {Inv}_R(\mathsf {read}_{\tau _0}(T))$ . Since ${i> j}$ and $\mathsf {read}_{\tau _0}(T)^{-1}(i) < \mathsf {read}_{\tau _0}(T)^{-1}(j)$ , the box $T^{-1}(j)$ is placed strictly left and weakly below $T^{-1}(i)$ . This, together with the definition of distinguished Schur labeling, implies that $\tau _{T^{-1}(i)} < \tau _{T^{-1}(j)}$ ; equivalently, $\mathsf {read}_{\tau }(T)^{-1}(i) < \mathsf {read}_{\tau }(T)^{-1}(j)$ . It follows that $(i,j) \in \mathrm {Inv}_R(\mathsf {read}_{\tau }(T))$ . Since we chose an arbitrary $(i,j) \in \mathrm {Inv}_R(\mathsf {read}_{\tau _0}(T))$ , we have the inclusion $\mathrm {Inv}_R(\mathsf {read}_{\tau _0}(T)) \subseteq \mathrm {Inv}_R(\mathsf {read}_{\tau }(T))$ . Therefore, $\mathsf {read}_{\tau _0}(T) \preceq _R \mathsf {read}_{\tau }(T)$ .

Let us show the opposite inclusion $\{\mathsf {read}_\tau (T) \mid \tau \in \mathsf {DS}(\lambda /\mu )\} \supseteq [\mathsf {read}_{\tau _0}(T), \mathsf {read}_{\tau _1}(T)]_R$ . Since $\{\mathsf {read}_\tau (T) \mid \tau \text {is a bijective tableau of shape} \lambda / \mu \}$ is equal to $\mathfrak {S}_n$ as a set, the inclusion can be obtained by proving that $\mathsf {read}_\tau (T) \notin [\mathsf {read}_{\tau _0}(T), \mathsf {read}_{\tau _1}(T)]_R$ for all bijective tableaux $\tau $ of shape $\lambda /\mu $ with $\tau \notin \mathsf {DS}(\lambda /\mu )$ . To prove it, choose an arbitrary bijective tableau $\tau $ of shape $\lambda /\mu $ with $\tau \notin \mathsf {DS}(\lambda /\mu )$ . Since $\tau \notin \mathsf {DS}(\lambda /\mu )$ , there exists $ 1 \leq i < j \leq n$ such that i is weakly below and weakly left of j in $\tau $ . Set $x := T_{\tau ^{-1}(i)}$ and $y := T_{\tau ^{-1}(j)}$ . Then, x appears left of y in $\mathsf {read}_\tau (T)$ . However, since $\tau _0, \tau _1 \in \mathsf {DS}(\lambda /\mu )$ , we have $(\tau _0)_{\tau ^{-1}(i)}> (\tau _0)_{\tau ^{-1}(j)}$ and $(\tau _1)_{\tau ^{-1}(i)}> (\tau _1)_{\tau ^{-1}(j)}$ , which implies that x appears right of y in both $\mathsf {read}_{\tau _0}(T)$ and $\mathsf {read}_{\tau _1}(T)$ . If $x < y$ , then $(y, x) \in \mathrm {Inv}_R(\mathsf {read}_{\tau _0}(T))$ and $(y, x) \notin \mathrm {Inv}_R(\mathsf {read}_\tau (T))$ , thus $\mathrm {Inv}_R(\mathsf {read}_{\tau _0}(T)) \not \subseteq \mathrm {Inv}_R(\mathsf {read}_{\tau }(T))$ . Similarly, if $x> y$ , then $\mathrm {Inv}_R(\mathsf {read}_\tau (T)) \not \subseteq \mathrm {Inv}_R(\mathsf {read}_{\tau _1}(T))$ . Hence, $\mathsf {read}_\tau (T) \notin [\mathsf {read}_{\tau _0}(T), \mathsf {read}_{\tau _1}(T)]_R$ .

As a last step, we define two specific standard Young tableaux. For a skew partition $\lambda /\mu $ of size n, let $T_{\lambda /\mu }$ (resp. $T^{\prime}_{\lambda /\mu }$ ) be the standard Young tableau obtained by filling $\mathtt {yd}(\lambda /\mu )$ by $1, 2, \ldots , n$ from left to right starting with the top row (resp. from top to bottom starting with leftmost column).

Example 3.8. Let $\lambda / \mu = (2,2) \star (3,2) / (1)$ . Then

Now, we are ready prove the main theorem of this section.

Theorem 3.9. Let $P \in \mathsf {RSP}_n$ and $\lambda /\mu = \mathrm {sh}(\tau _P)$ . Then

$$\begin{align*}\Sigma_L(P) = [\mathsf{read}_{\tau_P}(T_{\lambda/\mu}), \mathsf{read}_{\tau_P}(T^{\prime}_{\lambda/\mu})]_L. \end{align*}$$

Proof. Due to Theorem 2.4 and Lemma 3.2, it suffices to show that $\mathsf {read}_{\tau _P}(T_{\lambda /\mu })$ is minimal and $\mathsf {read}_{\tau _P}(T^{\prime}_{\lambda /\mu })$ is maximal in $\mathsf {read}_{\tau _P}(\mathrm {SYT}(\lambda /\mu ))$ with respect to $\preceq _L$ . Let $T \in \mathrm {SYT}(\lambda / \mu )$ . In the case where $\tau _P = \tau _0$ , one can easily see that

(3.4) $$ \begin{align} \mathrm{Inv}_L(\mathsf{read}_{\tau_0}(T_{\lambda/\mu})) \subseteq \mathrm{Inv}_L(\mathsf{read}_{\tau_0}(T)) \subseteq \mathrm{Inv}_L(\mathsf{read}_{\tau_0}(T_{\lambda/\mu}')). \end{align} $$

In the case where $\tau _P \neq \tau _0$ , we consider the equality

(3.5) $$ \begin{align} \mathsf{read}_{\tau_P}(T) = \mathsf{read}_{\tau_0}(T) \mathsf{read}_{\tau_0}(\tau_P)^{-1} \end{align} $$

which follows from Definition 3.1. By Lemma 3.6, we have $\tau _P \in \mathsf {DS}(\lambda /\mu )$ ; therefore, combining Lemma 3.7 with Equation (3.5) yields that

$$ \begin{align*} \ell(\mathsf{read}_{\tau_P}(T)) = \ell(\mathsf{read}_{\tau_0}(T)) + \ell(\mathsf{read}_{\tau_0}(\tau_P)^{-1}). \end{align*} $$

Now, we have that

$$ \begin{align*} \ell(\mathsf{read}_{\tau_P}(T)) - \ell(\mathsf{read}_{\tau_P}(T_{\lambda/\mu})) & = \ell(\mathsf{read}_{\tau_0}(T)) - \ell(\mathsf{read}_{\tau_0}(T_{\lambda/\mu})) \\ & = \ell(\mathsf{read}_{\tau_0}(T) \mathsf{read}_{\tau_0}(T_{\lambda/\mu})^{-1}) & \text{by Equation }({3.4}) \\ & = \ell(\mathsf{read}_{\tau_P}(T) \mathsf{read}_{\tau_P}(T_{\lambda/\mu})^{-1}) & \text{by Definition }{3.1}. \end{align*} $$

Therefore, $\mathsf {read}_{\tau _P}(T_{\lambda /\mu }) \preceq _L \mathsf {read}_{\tau _P}(T)$ . In the same manner, we can prove that $\mathsf {read}_{\tau _P}(T) \preceq _L \mathsf {read}_{\tau _P}(T^{\prime}_{\lambda /\mu })$ .

4 An equivalence relation on $\mathrm {Int}(n)$

Recall that $\mathrm {Int}(n)$ denotes the set of nonempty left weak Bruhat intervals in $\mathfrak {S}_n$ ; that is,

$$ \begin{align*}\mathrm{Int}(n) = \{[\sigma,\rho]_L \mid \sigma,\rho \in \mathfrak{S}_n \text{ and } \sigma \preceq_L \rho\}. \end{align*} $$

For $I_1, I_2 \in \mathrm {Int}(n)$ , a poset isomorphism $f: (I_1, \preceq _L) \rightarrow (I_2, \preceq _L)$ is called descent-preserving if $\mathrm {Des}_L(\gamma ) = \mathrm {Des}_L(f(\gamma ))$ for all $\gamma \in I_1$ . In this section, we study the classification of left weak Bruhat intervals in $\mathrm {Int}(n)$ up to descent-preserving poset isomorphism. In particular, in the case where ${P \in \mathsf {RSP}_n}$ , we explicitly describe the isomorphism class of $\Sigma _L(P)$ .

We begin by explaining the reason why we consider descent-preserving poset isomorphisms. Note that every interval $I \in \mathrm {Int}(n)$ can be represented by the colored digraph whose vertices are given by the permutations in I and $\{1,2,\ldots , n-1\}$ -colored arrows are given by

$$ \begin{align*} \gamma \overset{i}{\rightarrow} \gamma' \quad \text{if and only if} \quad \gamma \preceq_L \gamma'\text{ and }s_i \gamma = \gamma'. \end{align*} $$

For intervals $I_1, I_2 \in \mathrm {Int}(n)$ , a map $f: I_1 \rightarrow I_2$ is called a colored digraph isomorphism if f is bijective and satisfies that for all $\gamma , \gamma ' \in I_1$ and $1 \leq i \leq n-1$ ,

$$ \begin{align*} \gamma \overset{i}{\rightarrow} \gamma' \quad \text{if and only if} \quad f(\gamma) \overset{i}{\rightarrow} f(\gamma'). \end{align*} $$

If there exists a descent-preserving colored digraph isomorphism between two intervals $I_1$ and $I_2$ , then $\mathsf {B}(I_1)$ is isomorphic to $\mathsf {B}(I_2)$ . Motivated by this fact, in [19, Subsection 3.1], the authors posed the classification problem of weak Bruhat intervals up to descent-preserving colored digraph isomorphism.

A colored digraph isomorphism between $I_1$ and $I_2$ is a poset isomorphism with respect to $\preceq _L$ , but a poset isomorphism $f: I_1 \rightarrow I_2$ may not be a colored digraph isomorphism. For instance, the poset isomorphism $f: [1234, 2134]_L \rightarrow [1234, 1324]_L$ defined by $f(1234) = 1234$ and $f(2134) = 1324$ is not a colored digraph isomorphism since

However, if a poset isomorphism between left weak Bruhat intervals is descent-preserving, it indeed proves to be a colored digraph isomorphism.

Proposition 4.1. Let $I_1, I_2 \in \mathrm {Int}(n)$ . Every descent-preserving poset isomorphism $f: I_1 \rightarrow I_2$ is a colored digraph isomorphism.

Proof. Let $\gamma , \gamma ' \in I_1$ with $\gamma \preceq _L^c \gamma '$ . Since f is a poset isomorphism, we have $f(\gamma ) \preceq _L^c f(\gamma ')$ . Let $i,j \in [n-1]$ , satisfying $\gamma ' = s_i \gamma $ and $f(\gamma ') = s_j f(\gamma )$ . For the assertion, it suffices to show that $i = j$ . Let $D_1 := \mathrm {Des}_L(\gamma )$ and $D_2 := \mathrm {Des}_L(\gamma ')$ . Since $\gamma ' = s_i \gamma $ , we have

$$\begin{align*}\{i\} \subseteq (D_1 \cup D_2) \setminus (D_1 \cap D_2) \subseteq \{i-1, i, i+1\}. \end{align*}$$

In addition, since $D_1 = \mathrm {Des}_L(f(\gamma ))$ , $D_2 = \mathrm {Des}_L(f(\gamma '))$ and $f(\gamma ') = s_j f(\gamma )$ , we have $j \in D_2 \setminus D_1$ , and it follows that j is one of $i-1$ , i, and $i+1$ .

If $j = i-1$ , then $i-1, i \in D_2$ . It follows that

$$\begin{align*}\gamma' = \cdots \ i+1 \ \cdots \ i \ \cdots \ i-1 \ \cdots \quad \text{in one-line notation}; \end{align*}$$

equivalently,

$$\begin{align*}\gamma = \cdots \ i \ \cdots \ i+1 \ \cdots \ i-1 \ \cdots \quad \text{in one-line notation.} \end{align*}$$

This implies that $j=i-1 \in D_1$ , which is a contradiction to $j \notin \mathrm {Des}_L(f(\gamma ))$ . Therefore, $j\neq i-1$ .

In a similar manner, one can show that $j \neq i+1$ . Hence, $j = i$ , as required.

Proposition 4.1 says that classifying weak Bruhat intervals up to descent-preserving colored digraph isomorphism is equivalent to classifying weak Bruhat intervals up to descent-preserving poset isomorphism. With this equivalence in mind, we introduce an equivalence relation, whose reflexivity, symmetricity and transitivity are obvious.

Definition 4.2. We define an equivalence relation $\overset {D}{\simeq }$ on $\mathrm {Int}(n)$ by $I_1 \overset {D}{\simeq } I_2$ if there is a descent-preserving (poset) isomorphism between $(I_1, \preceq _L)$ and $(I_2, \preceq _L)$ .

For each equivalence class C, we define

$$\begin{align*}\xi_C := \rho \sigma^{-1} \quad \text{for any }[\sigma, \rho]_L \in C. \end{align*}$$

By Proposition 4.1, $\xi _C$ does not depend on the choice of $[\sigma , \rho ]_L \in C$ . We also define

$$ \begin{align*} \overline{\mathrm{min}}(C) := \{\sigma \mid [\sigma, \rho]_L \in C\} \quad \text{and} \quad \overline{\mathrm{max}}(C) := \{\rho \mid [\sigma, \rho]_L \in C \}. \end{align*} $$

From now on, we always regard $\overline {\mathrm {min}}(C)$ and $\overline {\mathrm {max}}(C)$ as subposets of $(\mathfrak {S}_n, \preceq _R)$ .

The following lemma is the initial step in the proof of the main result of this section (Theorem 4.6).

Lemma 4.3. Let C be an equivalence class under $\overset {D}{\simeq }$ with $\ell (\xi _C) = 1$ . Then, $\overline {\mathrm {min}}(C)$ is a right weak Bruhat interval in $(\mathfrak {S}_n, \preceq _R)$ .

Before proving the lemma, we provide an outline of the proof for the reader’s understanding. We first classify the equivalence classes under consideration according to the set X in Equation (4.2). Then, case by case, we show that $\overline {\mathrm {min}}(C)$ has a unique minimal element $\sigma _0$ . In particular, in Case 3, we introduce a specific permutation $\mathbf {w}_0$ and show that it is the unique minimal element of $\overline {\mathrm {min}}(C)$ . Using these results, we next show that $\overline {\mathrm {min}}(C)$ has the unique maximal element $\sigma _1$ . Finally, we show that $[\sigma _0, \sigma _1]_R \subseteq \overline {\mathrm {min}}(C)$ .

Proof. From the condition $\ell (\xi _C) = 1$ , it follows that $\xi _C = s_{i_0}$ for some $i_0 \in [n-1]$ .

First, let us prove that there exists a unique minimal element in $\overline {\mathrm {min}}(C)$ . Let

(4.1) $$ \begin{align} D_1 := \mathrm{Des}_L(\sigma) \quad \text{and} \quad D_2 := \mathrm{Des}_L(s_{i_0}\sigma) \quad \text{for any }[\sigma, s_{i_0}\sigma]_L \in C \end{align} $$

and

(4.2) $$ \begin{align} X := (D_1 \cup D_2) \setminus (D_1 \cap D_2). \end{align} $$

One sees that $\{i_0\} \subseteq X \subseteq \{i_0 - 1, i_0, i_0 + 1\}$ , and therefore, X can be one of the following:

$$\begin{align*}\{i_0-1, i_0, i_0+1\}, \ \ \{i_0\}, \ \ \{i_0-1, i_0\}, \ \ \text{and} \ \ \{i_0, i_0+1\}. \end{align*}$$

Case 1: $X = \{i_0 - 1, i_0, i_0 + 1\}$ . Since $i_0 - 1, i_0+1 \in D_1$ and $i_0 \notin D_1$ ,

$$\begin{align*}w_0(D_1) = \cdots \ i_0 \ i_0 - 1 \ \cdots \ i_0+2 \ i_0+1 \ \cdots \end{align*}$$

in one-line notation. Considering this equality, one can see that $[w_0(D_1), s_{i_0} w_0(D_1)]_L \in C$ . By Lemma 2.2, $w_0(D_1) \preceq _R \sigma $ for all $\sigma \in \mathfrak {S}_n$ with $\mathrm {Des}_L(\sigma ) = D_1$ . Thus, $w_0(D_1)$ is a unique minimal element in $\overline {\mathrm {min}}(C)$ .

Case 2: $X = \{i_0\}$ . In this case, we have

$$\begin{align*}w_0(D_2) = \cdots \ i_0 + 1 \ i_0 \cdots \ \quad \end{align*}$$

in one-line notation. Considering this equality, one can see that $[s_{i_0} w_0(D_2), w_0(D_2)]_L \in C$ . Again, by Lemma 2.2, $w_0(D_2) \preceq _R \sigma $ for all $\sigma \in \mathfrak {S}_n$ with $\mathrm {Des}_L(\sigma ) = D_2$ . Thus, $s_{i_0} w_0(D_2)$ is a unique minimal element in $\overline {\mathrm {min}}(C)$ .

Case 3: $X = \{i_0-1, i_0\}$ . When $i_0 + 1 \notin D_1$ , following the way as in Case 1, one can see that $w_0(D_1)$ is a unique minimal element in $\overline {\mathrm {min}}(C)$ .

From now on, assume that $i_0 + 1 \in D_1$ . We begin by introducing necessary notation. Let

$$\begin{align*}\mathsf{m}_1 := \min\{m \in [n - 1] \mid [m, i_0 - 1] \subseteq D_1 \} \quad \text{and} \quad \mathsf{m}_2 := \max\{m \in [n - 1] \mid [i_0 + 1, m] \subseteq D_1 \}. \end{align*}$$

And set

$$\begin{align*}p_1 := \mathsf{m}_1 - 1, \quad p_2 := \mathsf{m}_2 - i_0, \quad p_3 := i_0 - \mathsf{m}_1, \quad \text{and} \quad p_4 := n - (\mathsf{m}_2 + 1). \end{align*}$$

Let

• • $\mathbf {w}^{(1)}$ be the longest element of the subgroup $\mathfrak {S}_{D_1 \cap [p_1 - 1]}$ of $\mathfrak {S}_{p_1}$ ,

• • $\mathbf {w}^{(2)}$ be the longest element of $\mathfrak {S}_{p_2}$ ,

• • $\mathbf {w}^{(3)}$ be the longest element of $\mathfrak {S}_{p_3}$ , and

• • $\mathbf {w}^{(4)}$ be the longest element of the subgroup $\mathfrak {S}_{D_1 \cap [p_4 -1]}$ of $\mathfrak {S}_{p_4}$ .

With this notation, we define $\mathbf {w}_0$ to be the permutation given by

$$\begin{align*}\mathbf{w}_0(k) := \begin{cases} \mathbf{w}^{(1)}(k) & \text{if }k \in [p_1], \\ \mathbf{w}^{(2)}(k - p_1) + (i_0 + 1) & \text{if }k \in [p_1 +1, p_1 + p_2], \\ i_0 & \text{if }k = p_1 + p_2 + 1, \\ \mathbf{w}^{(3)}(k - (p_1 + p_2 + 1)) + (\mathsf{m}_1 - 1) & \text{if }k \in [p_1 + p_2 + 2, p_1 + p_2 + p_3 + 1], \\ i_0 + 1 & \text{if }k = p_1 + p_2 + p_3 + 2, \\ \mathbf{w}^{(4)}(k - (p_1 + p_2 + p_3 + 2)) + (\mathsf{m}_2 + 1) & \text{if }k \in [p_1 + p_2 + p_3 + 3, n]. \end{cases} \end{align*}$$

It should be remarked that

$$ \begin{align*} \mathbf{w}_0([1,p_1]) &= [1,\mathsf{m}_1-1], \\ \mathbf{w}_0([p_1+1,p_1+p_2]) & = [i_0+2,\mathsf{m}_2+1], \\ \mathbf{w}_0(p_1+p_2+1) &= i_0, \\ \mathbf{w}_0([p_1+p_2+2,p_1+p_2+p_3+1]) &= [\mathsf{m}_1, i_0-1], \\ \mathbf{w}_0(p_1+p_2+p_3+2) &= i_0+1, \quad \text{and} \\ \mathbf{w}_0([p_1+p_2+p_3+3,n]) &= [\mathsf{m}_2+2, n]. \end{align*} $$

From the definition of $\mathbf {w}_0$ , it follows that $[\mathbf {w}_0, s_{i_0} \mathbf {w}_0]_L \in C$ ; equivalently, $\mathbf {w}_0 \in \overline {\mathrm {min}}(C)$ . We claim that $\mathbf {w}_0$ is a unique minimal element in $\overline {\mathrm {min}}(C)$ . This can be verified by showing that every minimal element in $\overline {\mathrm {min}}(C)$ is equal to $\mathbf {w}_0$ . Let $\sigma _0$ be a minimal element in $\overline {\mathrm {min}}(C)$ . Set

$$ \begin{align*} \begin{array}{l} \mathcal{I}_{\mathrm{L}} := \{\sigma_0(k) \mid 1 \le k < \sigma_0^{-1}(i_0)\}, \\[.5ex] \mathcal{I}_{\mathrm{C}} := \{\sigma_0(k) \mid \sigma_0^{-1}(i_0) < k < \sigma_0^{-1}(i_0+1) \}, \\[.5ex] \mathcal{I}_{\mathrm{R}} := \{\sigma_0(k) \mid \sigma_0^{-1}(i_0 + 1) < k \le n\}. \end{array} \end{align*} $$

To begin with, we establish the equalities

(4.3) $$ \begin{align} \mathcal{I}_{\mathrm{L}} = [\mathsf{m}_1 -1] \cup [i_0+2, \mathsf{m}_2+1], \quad \mathcal{I}_{\mathrm{C}} = [\mathsf{m}_1, i_0 - 1], \quad \text{and} \quad \mathcal{I}_{\mathrm{R}} = [\mathsf{m}_2 + 2,n]. \end{align} $$

These equalities are derived by verifying the following claims.

Claim 1. $\mathcal {I}_{\mathrm {C}} = [\mathsf {m}_1, i_0 - 1]$ . Let us show $\mathcal {I}_{\mathrm {C}} \subseteq [\mathsf {m}_1, i_0 - 1]$ . First, to prove $\mathcal {I}_{\mathrm {C}} \subseteq [i_0 - 1]$ , we assume that there exists $i \in \mathcal {I}_{\mathrm {C}}$ such that $i \geq i_0$ . Let

$$ \begin{align*} k_1 & := \mathrm{max}\{k \in [n] \mid \sigma_0(k) \in \mathcal{I}_{\mathrm{C}} \ \text{and} \ \sigma_0(k) \ge i_0 \}. \end{align*} $$

By the definition of $\mathcal {I}_{\mathrm {C}}$ , $i_0,i_0+1 \notin \mathcal {I}_{\mathrm {C}}$ . If $i_0 + 2 \in \mathcal {I}_{\mathrm {C}}$ , then $i_0 + 1 \in D_1 \setminus D_2$ , which contradicts the assumption that $i_0 +1 \notin X$ . Since $\sigma _0(k_1) \in \mathcal {I}_{\mathrm {C}}$ , it follows that $\sigma _0(k_1) \geq i_0 + 3$ . And, by the choice of $k_1$ , we have $\sigma _0(k_1 + 1) \leq i_0+1$ . Putting these together yields that $[\sigma _0 s_{k_1}, s_{i_0} \sigma _0s_{k_1}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k_1} \prec _R \sigma _0$ . This contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ ; therefore, $\mathcal {I}_{\mathrm {C}} \subseteq [i_0 - 1]$ . Next, to prove $\mathcal {I}_{\mathrm {C}} \subseteq [\mathsf {m}_1, n]$ , we assume that there exists $i \in \mathcal {I}_{\mathrm {C}}$ such that $i < \mathsf {m}_1$ . Let

$$\begin{align*}k_2 := \mathrm{min}\{ k \in [n] \mid \sigma_0(k) \in \mathcal{I}_{\mathrm{C}} \ \text{and} \ \sigma_0(k) < \mathsf{m}_1\}. \end{align*}$$

By the choice of $k_2$ , we have $\sigma _0(k_2) + 1 \le \mathsf {m}_1 \le \sigma _0(k_2 - 1)$ . In addition, if $\sigma _0(k_2) + 1 = \mathsf {m}_1 = \sigma _0(k_2 - 1)$ , then $\mathsf {m}_1 - 1 \in \mathrm {Des}_L(\sigma _0)$ , which cannot happen by the definition of $\mathsf {m}_1$ . Therefore, $\sigma _0(k_2) + 1 < \sigma _0(k_2 - 1)$ , which implies that $[\sigma _0 s_{k_2 - 1}, s_{i_0} \sigma _0 s_{k_2 - 1}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k_2 - 1} \prec _R \sigma _0$ . This contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ . Thus, $\mathcal {I}_{\mathrm {C}} \subseteq [\mathsf {m}_1, i_0 - 1]$ .

Let us show $\mathcal {I}_{\mathrm {C}} \supseteq [\mathsf {m}_1, i_0 - 1]$ . Assume for the sake of contradiction that there exists $i \in [\mathsf {m}_1, i_0 - 1]$ such that $i \notin \mathcal {I}_{\mathrm {C}}$ . Let j be the maximal element in $[\mathsf {m}_1, i_0 - 1]$ such that $j \notin \mathcal {I}_{\mathrm {C}}$ . Since $i_0 - 1 \in X$ , we have $i_0 - 1 \in \mathcal {I}_{\mathrm {C}}$ . It follows that $j < i_0 - 1$ , so $j+1 \in [\mathsf {m}_1,i_0-1]$ . Combining this with the maximality of j, we have $j+1 \in \mathcal {I}_{\mathrm {C}}$ . And, by the definition of $\mathsf {m}_1$ , we have $j \in D_1$ . Putting these together yields that $j \in \mathcal {I}_{\mathrm {R}}$ . Let

$$\begin{align*}k_3 := \mathrm{min}\{ k \in [n] \mid \sigma_0(k) \in \mathcal{I}_{\mathrm{R}} \ \text{and} \ \sigma_0(k) \leq j \}. \end{align*}$$

If $\sigma _0(k_3 -1) \le \sigma _0(k_3) + 1$ , then $\sigma _0(k_3 -1) \le j + 1 < i_0$ . So, $\sigma _0(k_3 - 1) \neq i_0 + 1$ , which implies $\sigma _0(k_3 -1) \in \mathcal {I}_{\mathrm {R}}$ . This, together with the minimality of $k_3$ , yields that $j+1 \le \sigma _0(k_3 -1)$ . It follows that $\sigma _0(k_3 - 1) = j + 1$ , which is a contradiction because $\sigma _0(k_3 -1) \in \mathcal {I}_{\mathrm {R}}$ , but $j+1 \in \mathcal {I}_{\mathrm {C}}$ . Therefore, we have $\sigma _0(k_3) + 1 < \sigma _0(k_3-1)$ . In addition, since $j < i_0 - 1$ , we have $\sigma _0(k_3) < i_0 - 1$ . Putting these together yields that $[\sigma _0 s_{k_3 - 1}, s_{i_0} \sigma _0 s_{k_3 - 1}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k_3 - 1} \prec _R \sigma _0$ , which contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ . Thus, $\mathcal {I}_{\mathrm {C}} \supseteq [\mathsf {m}_1, i_0 - 1]$ .

Claim 2. $[\mathsf {m}_1 -1] \cup [i_0+2, \mathsf {m}_2+1] \subseteq \mathcal {I}_{\mathrm {L}}$ . By the definition of $\mathsf {m}_2$ , we have $[i_0 + 1, \mathsf {m}_2] \subseteq \mathrm {Des}_L(\sigma _0)$ . Since $i_0+2 \notin \mathcal {I}_{\mathrm {C}}$ , we have $[i_0+2, \mathsf {m}_2+1] \subseteq \mathcal {I}_{\mathrm {L}}$ . To prove $[\mathsf {m}_1 - 1] \subseteq \mathcal {I}_{\mathrm {L}}$ , suppose that there exists $i \in [\mathsf {m}_1 - 1]$ such that $i \notin \mathcal {I}_{\mathrm {L}}$ . Let

$$\begin{align*}k_4 := \mathrm{min}\{k \in [n] \mid \sigma_0(k) \in [\mathsf{m}_1 - 1] \ \text{and} \ \sigma_0(k) \notin \mathcal{I}_{\mathrm{L}}\}. \end{align*}$$

Since $\sigma _0(k_4) \notin \mathcal {I}_{\mathrm {L}}$ and $\sigma _0(k_4) < \mathsf {m}_1$ , we have $\sigma _0(k_4) \in \mathcal {I}_{\mathrm {R}}$ . This implies that $\sigma _0(k_4-1) \notin \mathcal {I}_{\mathrm {L}} \cup \{i_0\} \cup \mathcal {I}_{\mathrm {C}}$ . In addition, the minimality of $k_4$ gives $\sigma _0(k_4 - 1) \geq \mathsf {m}_1$ . Since $[\mathsf {m}_1, i_0-1] = \mathcal {I}_{\mathrm {C}}$ , we have $\sigma _0(k_4 - 1) \geq i_0+1$ . Putting the above inequalities together, we have

$$\begin{align*}\sigma_0(k_4) < \mathsf{m}_1 \le i_0-1 < i_0 + 1 \le \sigma_0(k_4-1), \end{align*}$$

and so $\sigma _0(k_4) + 2 < \sigma _0(k_4 - 1)$ . It follows that $[\sigma _0 s_{k_4 - 1}, s_{i_0} \sigma _0 s_{k_4 - 1}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k_4 - 1} \prec _R \sigma _0$ . This contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ ; thus, $[\mathsf {m}_1 - 1] \subseteq \mathcal {I}_{\mathrm {L}}$ .

Claim 3. $[\mathsf {m}_2 + 2,n] \subseteq \mathcal {I}_{\mathrm {R}}$ . Suppose that there exists $i \in [\mathsf {m}_2 + 2,n]$ such that $i \notin \mathcal {I}_{\mathrm {R}}$ . Let

$$\begin{align*}k_5 := \mathrm{max}\{k \in [n] \mid \sigma_0(k) \notin \mathcal{I}_{\mathrm{R}} \ \ \text{and} \ \ \sigma_0(k) \in [\mathsf{m}_2 + 2,n]\}. \end{align*}$$

Since $k_5 \notin \mathcal {I}_{\mathrm {R}}$ and $\sigma _0(k_5)> i_0 + 1$ , we have $\sigma _0(k_5) \in \mathcal {I}_{\mathrm {L}}$ , which implies that $\sigma _0(k_5 + 1) \notin \mathcal {I}_{\mathrm {R}}$ . By the maximality of $k_5$ , we have $\sigma _0(k_5+1) \le \mathsf {m}_2 + 1$ . If $\sigma _0(k_5+1) < \mathsf {m}_2 + 1$ , then

$$\begin{align*}\sigma_0(k_5+1) +1 < \mathsf{m}_2 + 2 \le \sigma_0(k_5). \end{align*}$$

If $\sigma _0(k_5+1) = \mathsf {m}_2 + 1$ , then $\sigma _0^{-1}(\mathsf {m}_2 + 2)> k_5+1$ due to the maximality of $\mathsf {m}_2$ , so $\sigma _0(k_5) \neq \mathsf {m}_2 + 2$ , which implies

$$\begin{align*}\sigma_0(k_5+1) + 1 < \sigma_0(k_5). \end{align*}$$

Putting these together with the inequalities $\sigma _0(k_5) \ge \mathsf {m}_2 + 2>i_0 + 2$ yields that $[\sigma _0 s_{k_5}, s_{i_0} \sigma _0 s_{k_5}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k_5} \prec _R \sigma _0$ . This contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ ; thus, $[\mathsf {m}_2 + 2, n] \subseteq \mathcal {I}_{\mathrm {R}}$ .

Now, we are ready to show that $\sigma _0 = \mathbf {w}_0$ . Let

$$\begin{align*}\mathcal{I}^{(1)}_{\mathrm{L}} := \{ \sigma_0(k) \in \mathcal{I}_{\mathrm{L}} \mid 1 \le k \le \mathsf{m}_1-1 \} \quad \text{and} \quad \mathcal{I}^{(2)}_{\mathrm{L}} := \{ \sigma_0(k) \in \mathcal{I}_{\mathrm{L}} \mid \mathsf{m}_1 \le k <\sigma_0^{-1}(i_0) \}. \end{align*}$$

We claim that $\mathcal {I}^{(1)}_{\mathrm {L}} = [\mathsf {m}_1-1]$ and $\mathcal {I}^{(2)}_{\mathrm {L}} = [i_0+2, \mathsf {m}_2+1]$ . We may assume that $\mathsf {m}_1> 1$ ; otherwise, the claim is obvious. To prove our claim, suppose that there exists $i \in \mathcal {I}^{(1)}_{\mathrm {L}}$ such that $i \in [i_0+2,\mathsf {m}_2+1]$ . Then, there exists $1 \le k < \sigma _0^{-1}(i_0) - 1$ such that $\sigma _0(k) \in [i_0+2, \mathsf {m}_2 + 1]$ and $\sigma _0(k + 1) \in [\mathsf {m}_1 - 1]$ . It follows that $[\sigma _0 s_{k}, s_{i_0} \sigma _0 s_{k}]_L \overset {D}{\simeq } [\sigma _0, s_{i_0}\sigma _0]_L$ and $\sigma _0 s_{k} \prec _R \sigma _0$ . Again, this contradicts the minimality of $\sigma _0$ in $\overline {\mathrm {min}}(C)$ , so

(4.4) $$ \begin{align} \mathcal{I}^{(1)}_{\mathrm{L}} = [\mathsf{m}_1-1] \quad \text{and} \quad \mathcal{I}^{(2)}_{\mathrm{L}} = [i_0+2, \mathsf{m}+1]. \end{align} $$

Putting Lemma 2.2, Equation (4.3), Equation (4.4) and the minimality of $\sigma _0$ together, we conclude that $\sigma _0 = \mathbf {w}_0$ .

Case 4: $X = \{i_0, i_0 + 1\}$ . Take $[\sigma , s_{i_0} \sigma ]_L \in C$ and let $C'$ be the equivalence class of $[\sigma ^{w_0}, (s_{i_0} \sigma )^{w_0}]_L$ . By mimicking Equation (4.1) and Equation (4.2), we define

$$\begin{align*}D^{\prime}_1 := \mathrm{Des}_L(\sigma^{w_0}), \quad D^{\prime}_2 := \mathrm{Des}_L((s_{i_0}\sigma)^{w_0}), \quad \text{and} \quad X' := (D^{\prime}_1 \cup D^{\prime}_2) \setminus (D^{\prime}_1 \cap D^{\prime}_2). \end{align*}$$

Since $D^{\prime}_1 = \{ n-i \mid i \in D_1\}$ and $D^{\prime}_2 = \{ n-i \mid i \in D_2\}$ , we have

$$\begin{align*}X' = \{n-i_0, (n-i_0)+1\}. \end{align*}$$

Following the proof of Case 3, we see that $\overline {\mathrm {min}}(C')$ has a unique minimal element $\mathbf {w}^{\prime}_0$ . And one can easily see that the map $f: \overline {\mathrm {min}}(C) \rightarrow \overline {\mathrm {min}}(C')$ , $\gamma \mapsto \gamma ^{w_0}$ is a well-defined bijection and that for $\gamma _1, \gamma _2 \in \overline {\mathrm {min}}(C)$ , $\gamma _1 \preceq _R \gamma _2$ if and only if $f(\gamma _1) \preceq _R f(\gamma _2)$ . Thus, $(\mathbf {w}^{\prime}_0)^{w_0}$ is a unique minimal element in $\overline {\mathrm {min}}(C')$ .

Second, we will show that $\overline {\mathrm {min}}(C)$ has a unique maximal element. Recall that we take $[\sigma , s_{i_0} \sigma ]_L \in C$ . Let $C"$ be the equivalence class of $[s_{i_0} \sigma w_0, \sigma w_0]_L$ . Due to the previous arguments, we know that there is a unique minimal element $\gamma _0$ in $\overline {\mathrm {min}}(C")$ . One can easily see that the map $g: \overline {\mathrm {min}}(C) \rightarrow \overline {\mathrm {min}}(C")$ , $\gamma \mapsto \gamma w_0$ is a well-defined bijection and that for $\gamma _1, \gamma _2 \in \overline {\mathrm {min}}(C)$ , $\gamma _1 \preceq _R \gamma _2$ if and only if $g(\gamma _1) \succeq _R g(\gamma _2)$ . Therefore, $\gamma _0 w_0$ is the unique maximal element in $\overline {\mathrm {min}}(C)$ .

Finally, we will show that $\overline {\mathrm {min}}(C)$ is a right weak Bruhat interval in $(\mathfrak {S}_n, \preceq _R)$ . Let $\sigma _0$ and $\sigma _1$ be the minimal and maximal elements in $\overline {\mathrm {min}}(C)$ , respectively. Let $\gamma \in [\sigma _0, \sigma _1]_R$ . Since $\mathrm {Des}_L(\sigma _0) = \mathrm {Des}_L(\sigma _1)$ , we have $\mathrm {Des}_L(\gamma ) = \mathrm {Des}_L(\sigma _0)$ by Lemma 2.2. Next, let us examine $\mathrm {Des}_L(s_{i_0}\gamma )$ . Since $\mathrm {Des}_L(\sigma _0) = \mathrm {Des}_L(\gamma )$ , it follows that $\gamma \preceq _L s_{i_0} \gamma $ . By Lemma 2.1, we have that $s_{i_0} \gamma \in [s_{i_0}\sigma _0, s_{i_0}\sigma _1]_{R}$ . Since $\mathrm {Des}_L(s_{i_0} \sigma _0) =\mathrm {Des}_L(s_{i_0} \sigma _1)$ , we have $\mathrm {Des}_L(s_{i_0}\gamma ) = \mathrm {Des}_L(s_{i_0}\sigma _0)$ by Lemma 2.2. Thus, ${\gamma \in \overline {\mathrm {min}}(C)}$ .

Example 4.4. Let $C \subseteq \mathrm {Int}(4)$ be the equivalence class of $[2134, 2143]_L$ . One sees that

$$\begin{align*}C = \{[2134, 2143]_L, [2314, 2413]_L, [2341, 2431]_L\}. \end{align*}$$

So, $\overline {\mathrm {min}}(C) = \{2134,2314,2341\}$ and $\overline {\mathrm {max}}(C) = \{2143,2413,2431\}$ which are equal to $[2134, 2341]_R$ and $[2143,2431]_R$ , respectively. For the readers’ convenience, we draw the left weak Bruhat intervals in C within the left weak Bruhat graph of $\mathfrak {S}_4$ on the left-hand side of Figure 1. We also draw the right weak Bruhat intervals $\overline {\mathrm {min}}(C)$ and $\overline {\mathrm {max}}(C)$ within the right weak Bruhat graph of $\mathfrak {S}_4$ on the right-hand side of Figure 1.

Figure 1 The left weak Bruhat intervals in C on $(\mathfrak {S}_4, \preceq _L)$ and the right weak Bruhat intervals $\overline {\mathrm {min}}(C)$ and $\overline {\mathrm {max}}(C)$ on $(\mathfrak {S}_4, \preceq _R)$ in Example 4.4.

Lemma 4.5. The intersection of two right weak Bruhat intervals in $\mathfrak {S}_n$ is again a right weak Bruhat interval.

Proof. It is well known that $(\mathfrak {S}_n, \preceq _R)$ is a lattice; that is, every two-element subset $\{\gamma _1,\gamma _2\} \subseteq \mathfrak {S}_n$ has the least upper bound and greatest lower bound (for example, see [6, Section 3.2]). Combining this with the fact $|\mathfrak {S}_n| < \infty $ , we derive the desired result.

The following theorem provides significant information regarding equivalence classes under $\overset {D}{\simeq }$ .

Theorem 4.6. Let C be an equivalence class under $\overset {D}{\simeq }$ . Then $\overline {\mathrm {min}}(C)$ and $\overline {\mathrm {max}}(C)$ are right weak Bruhat intervals in $(\mathfrak {S}_n, \preceq _R)$ .

Proof. Note that $\sigma \preceq _L \xi _C \sigma $ for any $\sigma \in \overline {\mathrm {min}}(C)$ and that $\overline {\mathrm {max}}(C) = \xi _C \cdot \overline {\mathrm {min}}(C)$ . If we prove that $\overline {\mathrm {min}}(C)$ is a right weak Bruhat interval, then Lemma 2.1 implies that $\overline {\mathrm {max}}(C)$ is also a right weak Bruhat interval. So we will only prove that $\overline {\mathrm {min}}(C)$ is a right weak Bruhat interval.

When $\ell (\xi _C) = 0$ , the assertion follows from Lemma 2.2. From now on, assume that $\ell (\xi _C) \geq 1$ . We will prove the assertion by using mathematical induction on $\ell (\xi _C)$ . When $\ell (\xi _C) = 1$ , the assertion is true by Lemma 4.3. Let k be an arbitrary positive integer and suppose that the assertion holds for every equivalence class $C \in \mathscr {C}(n)$ with $\ell (\xi _C) \leq k$ . Let $C \in \mathscr {C}(n)$ with $\ell (\xi _C) = k+1$ . Set

$$ \begin{align*}\mathcal{A} := \{i \in [n-1] \mid s_i \in [\mathrm{id}, \xi_C]_L \}. \end{align*} $$

Given $i \in \mathcal {A}$ and $\sigma \in \overline {\mathrm {min}}(C)$ , note that

$$ \begin{align*} [\sigma, s_i \sigma]_L \overset{D}{\simeq} [\sigma', s_i \sigma']_L \quad \text{and} \quad [s_i \sigma, \xi_C \sigma]_L \overset{D}{\simeq} [s_i\sigma', \xi_C \sigma']_L \end{align*} $$

for all $\sigma ' \in \overline {\mathrm {min}}(C)$ . This says that the equivalence classes of $[\sigma , s_i \sigma ]_L$ and $[s_i\sigma , \xi _C \sigma ]_L$ do not depend on $\sigma \in \overline {\mathrm {min}}(C)$ . For each $i \in \mathcal {A}$ , we set

$$ \begin{align*} E_i & := \text{the equivalence class of }[\sigma, s_i \sigma]_L, \\ E^{\prime}_i & := \text{the equivalence class of }[s_i \sigma, \xi_C \sigma]_L \end{align*} $$

for any $\sigma \in \overline {\mathrm {min}}(C)$ . Then, we set

$$ \begin{align*} J_i := \overline{\mathrm{max}}(E_i) \cap \overline{\mathrm{min}}(E^{\prime}_i) \quad \text{for }i \in \mathcal{A}, \quad \text{and} \quad J := \bigcap_{i \in \mathcal{A}}s_i \cdot J_i. \end{align*} $$

Now, the desired assertion can be achieved by proving the following claims:

1. (i) $\overline {\mathrm {min}}(C) = J$ .

2. (ii) J is a right weak Bruhat interval.

First, let us prove $\overline {\mathrm {min}}(C) = J$ . By the definition of $J_i$ , we have $s_i \sigma \in J_i$ for all $i \in \mathcal {A}$ and $\sigma \in \overline {\mathrm {min}}(C)$ . It follows that $\overline {\mathrm {min}}(C) \subseteq J$ . To prove the opposite inclusion $\overline {\mathrm {min}}(C) \supseteq J$ , take $\sigma \in J$ . By the definition of J, we have

$$ \begin{align*} [\sigma, s_i \sigma]_L \in E_i \quad \text{and} \quad [s_i \sigma ,\xi_C \sigma]_L \in E^{\prime}_i \quad \text{for all }i \in \mathcal{A}. \end{align*} $$

And Lemma 2.1 implies that

$$ \begin{align*}\{s_i \sigma \mid i \in \mathcal{A} \} = \{\gamma \in [\sigma, \xi_C\sigma]_L \mid \sigma \prec^{\mathrm{c}}_L \gamma \}. \end{align*} $$

Putting these together yields that $[\sigma , \xi _C \sigma ]_L \in C$ ; therefore, $\sigma \in \overline {\mathrm {min}}(C)$ .

Next, let us prove that J is a right weak Bruhat interval. Due to Lemma 4.5, it suffices to show that $s_i \cdot J_i$ is a right weak Bruhat interval for $i \in \mathcal {A}$ . Let us fix $i \in \mathcal {A}$ . Since $\ell (\xi _{E_i}) = 1$ and $\ell (\xi _{E^{\prime}_i}) = k$ , $\overline {\mathrm {max}}(E_i)$ and $\overline {\mathrm {min}}(E^{\prime}_i)$ are right weak Bruhat intervals by the induction hypothesis. Combining this with Lemma 4.5 yields that $J_i$ is a right weak Bruhat interval. In addition, we have $s_i \gamma \preceq _L \gamma $ for all $\gamma \in J_i$ . Therefore, by Lemma 2.1, $s_i \cdot J_i$ is a right weak Bruhat interval.

According to Theorem 4.6, every equivalence class C can be expressed as follows:

$$\begin{align*}C = \{[\gamma, \xi_C\gamma]_L \mid \gamma \in [\sigma_0, \sigma_1]_R\}, \end{align*}$$

where $\sigma _0$ and $\sigma _1$ represent the minimal and maximal elements in $\overline {\mathrm {min}}(C)$ , respectively. In particular, when C is the equivalence class of $\Sigma _L(P)$ for $P \in \mathsf {RSP}_n$ , we can provide an explicit description of it.

Theorem 4.7. Let $P \in \mathsf {RSP}_n$ and C the equivalence class of $\Sigma _L(P)$ under $\overset {D}{\simeq }$ . Then

$$\begin{align*}C = \{\Sigma_L(Q) \mid Q \in \mathsf{RSP}_n\text{ with }\mathrm{sh}(\tau_Q) = \mathrm{sh}(\tau_P) \}. \end{align*}$$

Proof. Let $\lambda /\mu = \mathrm {sh}(\tau _P)$ . Combining Equation (3.3) with Theorem 3.9 yields that

$$\begin{align*}\{\Sigma_L(Q) \mid Q \in \mathsf{RSP}_n\text{ with }\mathrm{sh}(\tau_Q) = \lambda / \mu \} = \{[\mathsf{read}_{\tau}(T_{\lambda/\mu}), \mathsf{read}_{\tau}(T^{\prime}_{\lambda/\mu})]_L \mid \tau \in \mathsf{DS}(\lambda/\mu) \}. \end{align*}$$

Therefore, for the assertion, we have only to show the equality

$$ \begin{align*} C = \{[\mathsf{read}_{\tau}(T_{\lambda/\mu}), \mathsf{read}_{\tau}(T^{\prime}_{\lambda/\mu})]_L \mid \tau \in \mathsf{DS}(\lambda/\mu) \}. \end{align*} $$

First, let us show that $\{[\mathsf {read}_{\tau }(T_{\lambda /\mu }), \mathsf {read}_{\tau }(T^{\prime}_{\lambda /\mu })]_L \mid \tau \in \mathsf {DS}(\lambda /\mu ) \} \subseteq C$ . This can be done by proving that for $\tau \in \mathsf {DS}(\lambda /\mu )$ , the map

$$ \begin{align*} f_{P;\tau}: [\mathsf{read}_{\tau_P}(T_{\lambda/\mu}), \mathsf{read}_{\tau_P}(T^{\prime}_{\lambda/\mu})]_L & \rightarrow [\mathsf{read}_{\tau}(T_{\lambda/\mu}), \mathsf{read}_{\tau}(T^{\prime}_{\lambda/\mu})]_L \\ \mathsf{read}_{\tau_P}(T) &\mapsto \mathsf{read}_{\tau}(T) \quad (T\in \mathrm{SYT}(\lambda / \mu)) \end{align*} $$

is a descent-preserving isomorphism. Let us fix $\tau \in \mathsf {DS}(\lambda /\mu )$ . The definition of $\tau $ -reading implies that for any $T_1,T_2 \in \mathrm {SYT}(\lambda /\mu )$ ,

$$ \begin{align*}\mathsf{read}_{\tau}(T_1) \preceq_L^{\mathrm{c}} \mathsf{read}_{\tau}(T_2) \quad \text{if and only if} \quad \mathsf{read}_{\tau_P}(T_1) \preceq_L^{\mathrm{c}} \mathsf{read}_{\tau_P}(T_2), \end{align*} $$

and therefore, $f_{P;\tau }$ is a poset isomorphism. To show that $f_{P;\tau }$ is descent-preserving, choose arbitrary $T \in \mathrm {SYT}(\lambda /\mu )$ and $i \in \mathrm {Des}_L(\mathsf {read}_{\tau _P}(T))$ . Combining the conditions $T \in \mathrm {SYT}(\lambda /\mu )$ and $\tau _P \in \mathsf {DS}(\lambda /\mu )$ with $i \in \mathrm {Des}_L(\mathsf {read}_{\tau _P}(T))$ yields that $i+1$ appears weakly above and strictly right of i in T. It follows that $i \in \mathrm {Des}_L(\mathsf {read}_{\tau }(T))$ , so $\mathrm {Des}_L(\mathsf {read}_{\tau _P}(T)) \subseteq \mathrm {Des}_L(\mathsf {read}_{\tau }(T))$ . In the same manner, one can show that $\mathrm {Des}_L(\mathsf {read}_{\tau }(T)) \subseteq \mathrm {Des}_L(\mathsf {read}_{\tau _P}(T))$ . Therefore, $f_{P;\tau }$ is a descent-preserving isomorphism.

Next, let us show $C \subseteq \{[\mathsf {read}_{\tau }(T_{\lambda /\mu }), \mathsf {read}_{\tau }(T^{\prime}_{\lambda /\mu })]_L \mid \tau \in \mathsf {DS}(\lambda /\mu ) \}$ . In the previous paragraph, we prove that $[\mathsf {read}_{\tau }(T_{\lambda /\mu }), \mathsf {read}_{\tau }(T^{\prime}_{\lambda /\mu })]_L \in C$ for any $\tau \in \mathsf {DS}(\lambda /\mu )$ . This implies that $\mathsf {read}_{\tau }(T^{\prime}_{\lambda /\mu }) = \xi _C \mathsf {read}_{\tau }(T_{\lambda /\mu })$ , and so it suffices to show that

$$ \begin{align*}\overline{\mathrm{min}}(C) \subseteq \{\mathsf{read}_\tau(T_{\lambda/\mu}) \mid \tau \in \mathsf{DS}(\lambda / \mu)\}. \end{align*} $$

Due to Lemma 3.7, this inclusion can be obtained by proving

$$ \begin{align*}\gamma \in [\mathsf{read}_{\tau_0}(T_{\lambda/\mu}), \mathsf{read}_{\tau_1}(T_{\lambda/\mu})]_R \quad \text{for any }\gamma \in \overline{\mathrm{min}}(C). \end{align*} $$

Let $\gamma \in \overline {\mathrm {min}}(C)$ . Since $\mathsf {read}_{\tau _0}(T_{\lambda /\mu }) \in \overline {\mathrm {min}}(C)$ , we have $\mathrm {Des}_L(\mathsf {read}_{\tau _0}(T_{\lambda /\mu })) = \mathrm {Des}_L(\gamma )$ . In addition, by the definitions of $\tau _0$ and $T_{\lambda /\mu }$ , we have $\mathsf {read}_{\tau _0}(T_{\lambda /\mu }) = w_0(\alpha ^{\mathrm {c}})$ , where $\alpha = (\lambda _1 - \mu _1, \lambda _2 - \mu _2, \ldots , \lambda _{\ell (\lambda )} - \mu _{\ell (\lambda )})$ . Putting these equalities together with Lemma 2.2 yields that $\mathsf {read}_{\tau _0}(T_{\lambda /\mu }) \preceq _{R} \gamma $ . Similarly, we have

$$\begin{align*}\mathrm{Des}_L(\mathsf{read}_{\tau_1}(T^{\prime}_{\lambda/\mu})) = \mathrm{Des}_L(\xi_C\gamma) \quad \text{and} \quad \mathsf{read}_{\tau_1}(T^{\prime}_{\lambda/\mu}) = w_0(\beta^{\mathrm{c}})w_0, \end{align*}$$

where $\beta = (\lambda ^{\mathrm {t}}_1 - \mu ^{\mathrm {t}}_1, \lambda ^{\mathrm {t}}_2 - \mu ^{\mathrm {t}}_2, \ldots , \lambda ^{\mathrm {t}}_{\ell (\lambda ^{\mathrm {t}})} - \mu ^{\mathrm {t}}_{\ell (\lambda ^{\mathrm {t}})})$ . This, together with Lemma 2.2, yields that $\xi _C\gamma \preceq _{R} \mathsf {read}_{\tau _1}(T^{\prime}_{\lambda /\mu })$ . Since $\mathsf {read}_{\tau _1}(T_{\lambda /\mu }) \preceq _L \mathsf {read}_{\tau _1}(T^{\prime}_{\lambda /\mu }) = \xi _C \mathsf {read}_{\tau _1}(T_{\lambda /\mu })$ , we have $\gamma \preceq _{R} \mathsf {read}_{\tau _1}(T_{\lambda /\mu })$ . Therefore, $\gamma \in [\mathsf {read}_{\tau _0}(T_{\lambda /\mu }), \mathsf {read}_{\tau _1}(T_{\lambda /\mu })]_R$ , as desired.

Theorem 4.7 tells us that $\{\Sigma _L(P) \mid P \in \mathsf {RSP}_n \}$ is closed under $\overset {D}{\simeq }$ and the equivalence classes inside it are parametrized by the skew partitions of size n. Given a skew partition $\lambda /\mu $ of size n, let $C_{\lambda /\mu }$ be the equivalence class parametrized by $\lambda /\mu $ ; that is,

$$\begin{align*}C_{\lambda/\mu} = \{\Sigma_L(P) \mid P \in \mathsf{RSP}_n\text{ with }\mathrm{sh}(\tau_P)= \lambda/\mu \}. \end{align*}$$

Corollary 4.8. With the above notation, we have

$$\begin{align*}\{\Sigma_L(P) \mid P \in \mathsf{RSP}_n \} = \bigsqcup_{|\lambda/\mu| = n} C_{\lambda / \mu} \quad (\text{disjoint union}). \end{align*}$$

5 The classification of $\mathsf {M}_P$ ’s for $P \in \mathsf {RSP}_n$

Let $P, Q \in \mathsf {RSP}_n$ . By combining Proposition 4.1 with Theorem 4.7, we can see that if $\tau _P$ and $\tau _Q$ have the same shape, then the $H_n(0)$ -modules $\mathsf {M}_P$ and $\mathsf {M}_Q$ are isomorphic. The purpose of this section is to demonstrate that the converse of this implication also holds. Let us briefly explain our strategy. First, we provide both a projective cover and an injective hull of $\mathsf {M}_P$ for every $P \in \mathsf {RSP}_n$ . We discover that these modules are completely determined by the shape of $\tau _P$ , as demonstrated in Lemma 5.4. Then, we establish that if $\tau _P$ and $\tau _Q$ have different shapes, $\mathsf {M}_P$ and $\mathsf {M}_Q$ have either nonisomorphic projective covers or nonisomorphic injective hulls, as proven in Theorem 5.5.

To begin with, we present a brief overview of the background knowledge concerning projective modules and injective modules of the $0$ -Hecke algebras. In [11, Proposition 4.1], it was shown that $H_n(0)$ is a Frobenius algebra. It is well known that every Frobenius algebra is self injective, and for a finitely generated module M of a self injective algebra, M is projective if and only if it is injective (for instance, see [3, Proposition 1.6.2]). In [30], a complete list of non-isomorphic projective indecomposable $H_n(0)$ -modules was provided.

In the work [19], it was shown that this list can also be expressed in terms of weak Bruhat interval modules, specifically as $\{\mathbf {P}_\alpha \mid \alpha \models n\}$ , where

$$\begin{align*}\mathbf{P}_\alpha := \mathsf{B}(w_0(\alpha^{\mathrm{c}}), w_0 w_0(\alpha)) \quad \text{for }\alpha \models n. \end{align*}$$

We note that $\mathbf {P}_\alpha / \mathrm {rad} \; \mathbf {P}_\alpha $ is isomorphic to $\mathbf {F}_\alpha $ , where $\mathrm {rad} \; \mathbf {P}_\alpha $ is the radical of $\mathbf {P}_\alpha $ , the intersection of maximal submodules of $\mathbf {P}_\alpha $ .

In the following, we recall the definition of a projective cover and an injective hull. Let M be a finitely generated $H_n(0)$ -module. A projective cover of M is a pair $(\boldsymbol {P}, f)$ consisting of a projective $H_n(0)$ -module $\boldsymbol {P}$ and an $H_n(0)$ -module epimorphism $f: \boldsymbol {P} \rightarrow M$ such that $\ker (f) \subseteq \mathrm {rad}(\boldsymbol {P})$ . An injective hull of M is a pair $(\boldsymbol {I}, \iota )$ , where $\boldsymbol {I}$ is an injective $H_n(0)$ -module and $\iota : M \rightarrow \boldsymbol {I}$ is an $H_n(0)$ -module monomorphism satisfying $\iota (M) \supseteq \mathrm {soc}(\boldsymbol {I})$ . Here, $\mathrm {soc}(\boldsymbol {I})$ is the socle of $\boldsymbol {I}$ , the sum of all irreducible submodules of $\boldsymbol {I}$ . A projective cover and an injective hull of M always exist, and they are unique up to isomorphism. For more information, refer to [1, 23].

The projective modules introduced by Huang [18] play an important role in describing the projective cover and injective hull of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ . We briefly review these projective modules from the viewpoint of weak Bruhat interval modules. A generalized composition ${\boldsymbol {\unicode{x3b1} }}$ of n is a formal expression $\alpha ^{(1)} \star \alpha ^{(2)} \star \cdots \star \alpha ^{(k)}$ , where $\alpha ^{(i)} \models n_i$ for positive integers $n_i$ ’s with $n_1 + n_2 + \cdots + n_k = n$ . For compositions $\alpha = (\alpha _1, \alpha _2, \ldots , \alpha _{\ell (\alpha )})$ and $\beta = (\beta _1, \beta _2, \ldots , \beta _{\ell (\beta )})$ , let

$$ \begin{align*}\alpha \cdot \beta = (\alpha_1, \alpha_2, \ldots, \alpha_{\ell(\alpha)}, \beta_1, \beta_2, \ldots, \beta_{\ell(\beta)}) \quad \text{and} \quad \alpha \odot \beta = (\alpha_1, \alpha_2, \ldots, \alpha_{\ell(\alpha)}+\beta_1, \beta_2, \ldots, \beta_{\ell(\beta)}). \end{align*} $$

For a generalized composition ${\boldsymbol {\unicode{x3b1} }} = \alpha ^{(1)} \star \alpha ^{(2)} \star \cdots \star \alpha ^{(k)}$ , let

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}_{\bullet} := \alpha^{(1)} \cdot \alpha^{(2)} \cdot \cdots \cdot \alpha^{(k)}, \quad {\boldsymbol{\unicode{x3b1}}}_{\odot} := \alpha^{(1)} \odot \alpha^{(2)} \odot \cdots \odot \alpha^{(k)} \end{align*}$$

and let

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}^{\mathrm{c}} := (\alpha^{(1)})^{\mathrm{c}} \star (\alpha^{(2)})^{\mathrm{c}} \star \cdots \star (\alpha^{(k)})^{\mathrm{c}}, \quad {\boldsymbol{\unicode{x3b1}}}^{\mathrm{r}} := (\alpha^{(k)})^{\mathrm{r}} \star (\alpha^{(k-1)})^{\mathrm{r}} \star \cdots \star (\alpha^{(1)})^{\mathrm{r}}, \end{align*}$$

and ${\boldsymbol {\unicode{x3b1} }}^{\mathrm {c} \cdot \mathrm {r}} := ({\boldsymbol {\unicode{x3b1} }}^{\mathrm {c}})^{\mathrm {r}}$ . Normally, $({\boldsymbol {\unicode{x3b1} }}_{\bullet })^{\mathrm {c}} \neq ({\boldsymbol {\unicode{x3b1} }}^{\mathrm {c}})_{\bullet }$ and $({\boldsymbol {\unicode{x3b1} }}_{\odot })^{\mathrm {c}} \neq ({\boldsymbol {\unicode{x3b1} }}^{\mathrm {c}})_{\odot }$ for a generalized composition ${\boldsymbol {\unicode{x3b1} }}$ . Despite the potential for confusion, for the sake of brevity, we denote $({\boldsymbol {\unicode{x3b1} }}_{\bullet })^{\mathrm {c}}$ and $({\boldsymbol {\unicode{x3b1} }}_{\odot })^{\mathrm {c}}$ as ${\boldsymbol {\unicode{x3b1} }}_{\bullet }^{\mathrm {c}}$ and ${\boldsymbol {\unicode{x3b1} }}_{\odot }^{\mathrm {c}}$ , respectively. Then, we define

$$ \begin{align*}\mathbf{P}_{\boldsymbol{\unicode{x3b1}}} := \mathsf{B}(w_0({\boldsymbol{\unicode{x3b1}}}_{\bullet}^{\mathrm{c}}), w_0w_0({\boldsymbol{\unicode{x3b1}}}_{\odot})). \end{align*} $$

Huang decomposed $\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}}$ into projective indecomposable modules, and thus showed that it is projective. To be precise, the following lemma was shown.

Lemma 5.1 [18, Theorem 3.3]

For a generalized composition ${\boldsymbol {\unicode{x3b1} }} = \alpha ^{(1)} \star \alpha ^{(2)} \star \cdots \star \alpha ^{(k)}$ of n,

$$\begin{align*}\mathbf{P}_{\boldsymbol{\unicode{x3b1}}} \cong \mathbf{P}_{\alpha^{(1)}} \boxtimes \mathbf{P}_{\alpha^{(2)}} \boxtimes \cdots \boxtimes \mathbf{P}_{\alpha^{(k)}} \cong \bigoplus_{\beta \in [{\boldsymbol{\unicode{x3b1}}}]} \mathbf{P}_\beta, \end{align*}$$

where $[{\boldsymbol {\unicode{x3b1} }}] := \{\alpha ^{(1)} \; \square \; \alpha ^{(2)} \; \square \; \cdots \; \square \; \alpha ^{(k)} \mid \square = \cdot \; \text {or} \; \odot \}$ .

It is clear from Lemma 5.1 that if ${\boldsymbol {\unicode{x3b1} }}$ and ${\boldsymbol {\unicode{x3b2} }}$ are distinct generalized compositions of n, then $\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}$ and $\mathbf {P}_{\boldsymbol {\unicode{x3b2} }}$ are nonisomorphic. Let ${\boldsymbol {\unicode{x3b1} }}$ be a generalized composition of n. For $\rho \in [w_0({\boldsymbol {\unicode{x3b1} }}_\odot ^{\mathrm {c}}), w_0 w_0({\boldsymbol {\unicode{x3b1} }}_\odot )]_L$ , let $\Upsilon _{{\boldsymbol {\unicode{x3b1} }};\rho }: \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}} \rightarrow \mathsf {B}(w_0({\boldsymbol {\unicode{x3b1} }}_\bullet ^{\mathrm {c}}), \rho )$ be a $\mathbb C$ -linear map given by

$$\begin{align*}\gamma \mapsto\begin{cases}\gamma & \text{if }\gamma \in [w_0({\boldsymbol{\unicode{x3b1}}}_\bullet^{\mathrm{c}}), \rho]_L,\\0 & \text{if }\gamma \in [w_0({\boldsymbol{\unicode{x3b1}}}_\bullet^{\mathrm{c}}), w_0 w_0({\boldsymbol{\unicode{x3b1}}}_\odot)]_L \setminus [w_0({\boldsymbol{\unicode{x3b1}}}_\bullet^{\mathrm{c}}), \rho]_L.\end{cases} \end{align*}$$

Clearly, $\Upsilon _{{\boldsymbol {\unicode{x3b1} }};\rho }$ is an $H_n(0)$ -module epimorphism. In addition, it follows from [20, Lemma 6.2] that $\ker (\Upsilon _{{\boldsymbol {\unicode{x3b1} }};\rho }) \subseteq \mathrm {rad} (\mathbf {P}_{\boldsymbol {\unicode{x3b1} }})$ . Consequently, we have the following lemma.

Lemma 5.2. [(cf. [20, Lemma 6.2])] For a generalized composition ${\boldsymbol {\unicode{x3b1} }}$ of n and $\rho \in [w_0({\boldsymbol {\unicode{x3b1} }}_\odot ^{\mathrm {c}}), w_0 w_0({\boldsymbol {\unicode{x3b1} }}_\odot )]_L$ , the pair $(\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}, \Upsilon _{{\boldsymbol {\unicode{x3b1} }};\rho })$ is a projective cover of $\mathsf {B}(w_0({\boldsymbol {\unicode{x3b1} }}_\bullet ^{\mathrm {c}}), \rho )$ .

Let us provide notation and a lemma needed to describe a projective cover and an injective hull of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ . For a connected skew partition $\lambda / \mu $ of size n, define

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda/\mu) := (\lambda_1 - \mu_1, \lambda_2 - \mu_2, \ldots , \lambda_{\ell(\lambda)} - \mu_{\ell(\lambda)}). \end{align*}$$

And, for a disconnected skew partition $\lambda / \mu $ of size n, define

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda/\mu) := {\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda^{(1)} / \mu^{(1)}) \star {\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda^{(2)} / \mu^{(2)}) \star \cdots \star {\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda^{(k)} / \mu^{(k)}), \end{align*}$$

where $\lambda /\mu = \lambda ^{(1)} / \mu ^{(1)} \star \lambda ^{(2)} / \mu ^{(2)} \star \cdots \star \lambda ^{(k)} / \mu ^{(k)}$ with connected skew partitions $\lambda ^{(i)} / \mu ^{(i)}$ ’s $(1\le i \le k)$ .

Lemma 5.3. Let $\lambda /\mu $ be a skew partition of size n.

1. (1) $\mathsf {read}_{\tau _0}(T_{\lambda /\mu }) = w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )_{\bullet }^{\mathrm {c}})$ .

2. (2) $\mathsf {read}_{\tau _0}(T^{\prime}_{\lambda /\mu }) \in [w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )_{\odot }^{\mathrm {c}}), w_0w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )_{\odot })]_L$ .

Proof. By the definition of ${\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )$ , the assertion (1) is clear. In addition, one can easily see that $\mathrm {Des}_R(\mathsf {read}_{\tau _0}(T^{\prime}_{\lambda /\mu })) = \mathrm {set}({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )_{\odot }^{\mathrm {c}})$ . So, by Lemma 2.2, the assertion (2) follows.

Let $P \in \mathsf {RSP}_n$ and $\lambda /\mu = \mathrm {sh}(\tau _P)$ . By Theorem 3.9, $\mathsf {M}_P = \mathsf {B}(\mathsf {read}_{\tau _P}(T_{\lambda /\mu }), \mathsf {read}_{\tau _P}(T^{\prime}_{\lambda /\mu }))$ . Furthermore, by Theorem 4.7, we have an $H_n(0)$ -module isomorphism

$$ \begin{align*}f_{P}: \mathsf{M}_{\mathsf{poset}(\tau_0)} \rightarrow \mathsf{M}_P, \ \ \mathsf{read}_{\tau_0}(T) \mapsto \mathsf{read}_{\tau_P}(T) \;\;(T \in \mathrm{SYT}(\lambda/\mu)). \end{align*} $$

Set

$$ \begin{align*}\eta_P := f_P \circ \Upsilon_{{\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda/\mu); \mathsf{read}_{\tau_0}(T^{\prime}_{\lambda/\mu})}. \end{align*} $$

Combining Lemma 5.2 and Lemma 5.3 implies that the pair $\left (\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )}, \eta _P\right )$ is a projective cover of $\mathsf {M}_P$ .

To find an injective hull of $\mathsf {M}_P$ , we note that

$$\begin{align*}\mathsf{read}_{\tau_1}(T_{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}}) w_0 = \mathsf{read}_{\tau_0}(T^{\prime}_{\lambda/\mu}) \quad \text{and} \quad \mathsf{read}_{\tau_1}(T^{\prime}_{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}}) w_0 = \mathsf{read}_{\tau_0}(T_{\lambda/\mu}). \end{align*}$$

Combining these equalities with [19, Theorem 4] yields the following $H_n(0)$ -module isomorphism:

$$ \begin{align*} g_1: \mathbf{T}^{-}_{\widehat{\unicode{x3b8}}} \left( \mathsf{M}_{\mathsf{poset}(\tau_1^{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}})}\right) &\rightarrow \mathsf{M}_{\mathsf{poset}(\tau_0^{\lambda/\mu})}, \\ \gamma^* &\mapsto (-1)^{\ell(\gamma \; \mathsf{read}_{\tau_1}(T^{\prime}_{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}})^{-1})} \gamma w_0, \end{align*} $$

where $\gamma \in [\mathsf {read}_{\tau _1}(T_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}}), \mathsf {read}_{\tau _1}(T^{\prime}_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})]_L$ and $\gamma ^\ast $ denotes the dual of $\gamma $ with respect to the basis $[\mathsf {read}_{\tau _1}(T_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}}), \mathsf {read}_{\tau _1}(T^{\prime}_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})]_L$ for $\mathsf {M}_{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})}$ . Set

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}_{\mathrm{inj}}(\lambda/\mu) := {\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda^{\mathrm{t}} / \mu^{\mathrm{t}})^{\mathrm{c} \cdot \mathrm{r}}. \end{align*}$$

Again, by [19, Theorem 4], we have the $H_n(0)$ -module isomorphism

$$ \begin{align*} g_2: \mathbf{T}^{-}_{\widehat{\unicode{x3b8}}} \left( \mathbf{P}_{{\boldsymbol{\unicode{x3b1}}}_{\mathrm{proj}}(\lambda^{\mathrm{t}} / \mu^{\mathrm{t}})}\right) &\rightarrow \mathbf{P}_{{\boldsymbol{\unicode{x3b1}}}_{\mathrm{inj}}(\lambda/\mu)}, \\ \gamma^* &\mapsto (-1)^{\ell(\gamma (w_0w_0({\boldsymbol{\unicode{x3b1}}}_{\mathrm{inj}}(\lambda/\mu)_{\odot}))^{-1})} \gamma w_0, \end{align*} $$

where $\gamma \in [ w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}})_{\bullet }^{\mathrm {c}}), w_0w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}})_{\odot }) ]_L$ and $\gamma ^\ast $ denotes the dual of $\gamma $ with respect to the basis $[ w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}})_{\bullet }^{\mathrm {c}}), w_0w_0({\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}})_{\odot }) ]_L$ for $\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} / \mu ^{\mathrm {t}})}$ . Set $\eta _{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})} := f_{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})} \circ \Upsilon _{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}); \mathsf {read}_{\tau _0}(T^{\prime}_{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})}$ . As above, the pair $\left (\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} /\mu ^{\mathrm {t}})}, \eta _{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})}\right )$ is a projective cover of $\mathsf {M}_{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})}$ . And since $\mathbf {T}^{-}_{\widehat {\unicode{x3b8} }}$ is contravariant, $\left ( \mathbf {T}^{-}_{\widehat {\unicode{x3b8} }} \left ( \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{\mathrm {t}} /\mu ^{\mathrm {t}})} \right ), \mathbf {T}^{-}_{\widehat {\unicode{x3b8} }} \left ( \eta _{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})} \right ) \right )$ is an injective hull of $\mathbf {T}^{-}_{\widehat {\unicode{x3b8} }}(\mathsf {M}_{\mathsf {poset}(\tau _1^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}})})$ (for the definition of $\mathbf {T}^{-}_{\widehat {\unicode{x3b8} }} $ , see Section 2.4). Consequently, the pair $\left (\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\lambda /\mu )}, \iota _P\right )$ is an injective hull of $\mathsf {M}_P$ , where

$$\begin{align*}\iota_P = g_2 \circ \mathbf{T}^{-}_{\widehat{\unicode{x3b8}}} \left( \eta_{\mathsf{poset}(\tau_1^{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}})} \right) \circ g_1^{-1} \circ f_P^{-1}. \end{align*}$$

To summarize, we can state the following lemma.

Lemma 5.4. Let $P \in \mathsf {RSP}_n$ and $\lambda /\mu = \mathrm {sh}(\tau _P)$ .

1. (1) $\left ( \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )}, \eta _P \right )$ is a projective cover of $\mathsf {M}_P$ .

2. (2) $\left (\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\lambda /\mu )}, \iota _P \right )$ is an injective hull of $\mathsf {M}_P$ .

Now, we are ready to state the classification of $\mathsf {M}_P$ ’s for $P \in \mathsf {RSP}_n$ up to $H_n(0)$ -module isomorphism.

Theorem 5.5. Let $P, Q \in \mathsf {RSP}_n$ . Then

$$\begin{align*}\mathsf{M}_{P} \cong \mathsf{M}_{Q} \quad \text{if and only if} \quad \mathrm{sh}(\tau_P) = \mathrm{sh}(\tau_Q). \end{align*}$$

Proof. The ‘if’ part follows from Proposition 4.1 and Theorem 4.7. To prove the ‘only if’ part, suppose that $\mathsf {M}_{P} \cong \mathsf {M}_{Q}$ . For simplicity, let $\lambda /\mu = \mathrm {sh}(\tau _{P})$ and $\nu /\kappa = \mathrm {sh}(\tau _{Q})$ . By Lemma 5.4, $\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu )} \cong \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\nu /\kappa )}$ and $\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\lambda /\mu )} \cong \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\nu /\kappa )}$ , and therefore, ${\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu ) = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\nu /\kappa )$ and ${\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\lambda /\mu ) = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\nu /\kappa )$ . Since ${\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda /\mu ) = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\nu /\kappa )$ , the number of boxes in the same row of $\mathtt {yd}(\lambda /\mu )$ and $\mathtt {yd}(\nu /\kappa )$ are the same. Similarly, since ${\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\lambda /\mu ) = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {inj}}(\nu /\kappa )$ , the number of boxes in the same column of $\mathtt {yd}(\lambda /\mu )$ and $\mathtt {yd}(\nu /\kappa )$ are same. Thus, we have $\lambda /\mu = \nu /\kappa $ .

Note that Theorem 5.5 is the classification theorem concerning the class of $H_n(0)$ -modules $\{\mathsf {M}_P \mid P \in \mathsf {RSP}_n\}$ . Consequently, a natural question arises: can this theorem be extended to the classes $\{\mathsf {M}_P \mid P \in \mathsf {RP}_n\}$ or $\{\mathsf {M}_P \mid P \in \mathsf {SP}_n\}$ ? This question appears to be highly nontrivial, as it involves the investigation of a broader set of modules. As a specific instance, let us examine the characterization of posets $Q \in \mathsf {RP}_n$ such that $\mathsf {M}_Q \cong \mathsf {M}_P$ when $P \in \mathsf {RSP}_n$ . This problem can be readily addressed by assuming the validity of the following conjecture due to Stanley.

Conjecture 5.6 [34, p. 81]

For $P \in \mathsf {P}_n$ , if $K_P$ is symmetric, then $P \in \mathsf {SP}_n$ .

In more detail, by combining Stanley’s conjecture with Theorem 2.9(1), we can deduce that $\mathrm {ch}([\mathsf {M}_Q])$ is not symmetric, and as a consequence, $\mathsf {M}_Q \not \cong \mathsf {M}_P$ unless $Q \in \mathsf {SP}_n$ . This observation leads to the following conclusion from Theorem 5.5:

$$\begin{align*}\{Q \in \mathsf{RP}_n \mid \mathsf{M}_Q \cong \mathsf{M}_P\} = \{Q \in \mathsf{RSP}_n \mid \mathrm{sh}(\tau_P) = \mathrm{sh}(\tau_Q) \}. \end{align*}$$

If the shape of $\tau _P$ is non-skew, it is indeed possible to derive this conjectural identity without depending on the validity of Stanley’s conjecture (see Proposition 7.1). However, tackling the general case remains beyond our current comprehension. For further discussions on classifications, refer to Section 7.1.

6 A characterization of regular Schur labeled skew shape posets P and distinguished filtrations of $\mathsf {M}_P$

In this section, we prove that a poset $P \in \mathsf {P}_n$ is a regular Schur labeled skew shape poset if and only if $\Sigma _L(P)$ is dual plactic closed (Theorem 6.4). Then, by considering the dual plactic closedness of $\Sigma _L(P)$ , we construct filtrations

$$\begin{align*}0 =: M_0 \subsetneq M_1 \subsetneq M_2 \subsetneq \cdots \subsetneq M_l := \mathsf{M}_P \end{align*}$$

such that $\mathrm {ch}([M_{k}/M_{k-1}])$ is a Schur function for all $1 \leq k \leq l$ (Theorem 6.7).

6.1 A characterization of regular Schur labeled skew shape posets

Let $P \in \mathsf {P}_n$ and let $\Sigma _R(P) :=\{\gamma ^{-1} \mid \gamma \in \Sigma _L(P)\}$ . In [27, Fact 1], it was stated that if $P \in \mathsf {SP}_n$ , then $\Sigma _R(P)$ is plactic-closed. This, however, is not true. For instance, considering the case where $\lambda /\mu = (3,2)/(2)$ and , we have

$$\begin{align*}\Sigma_R(\mathsf{poset}(\tau)) = \{312, 231, 321\}, \end{align*}$$

which is not plactic-closed.

The purpose of this subsection is to prove that $P \in \mathsf {RSP}_n$ if and only if $\Sigma _R(P)$ is plactic-closed. We begin by providing background knowledge relevant to the plactic congruence. For instance, see [6, 14, 31, 35].

For $\sigma \in \mathfrak {S}_n$ and $1 < i < n$ , we write $\sigma \overset {1}{\cong } \sigma s_i$ if

$$\begin{align*}\sigma(i) < \sigma(i-1) < \sigma(i+1) \quad \text{or} \quad \sigma(i+1) < \sigma(i-1) < \sigma(i). \end{align*}$$

And we write $\sigma \overset {2}{\cong } \sigma s_{i-1}$ if

$$\begin{align*}\sigma(i-1) < \sigma(i+1) < \sigma(i) \quad \text{or} \quad \sigma(i) < \sigma(i+1) < \sigma(i-1). \end{align*}$$

The Knuth equivalence (or plactic congruence) is an equivalence relation $\overset {K}{\cong }$ on $\mathfrak {S}_n$ defined by $\sigma \overset {K}{\cong } \rho $ if and only if there are $\gamma _1,\gamma _2, \ldots , \gamma _k \in \mathfrak {S}_n$ such that

$$\begin{align*}\sigma = \gamma_1 \overset{a_1}{\cong} \gamma_2 \overset{a_2}{\cong} \cdots \overset{a_{k-1}}{\cong} \gamma_k = \rho, \end{align*}$$

where $a_1,a_2,\ldots a_{k-1} \in \{1,2\}$ . A subset S of $\mathfrak {S}_n$ is called plactic-closed if for any $\sigma \in S$ , every $\rho \in \mathfrak {S}_n$ with $\rho \overset {K}{\cong } \sigma $ is also an element of S; in other words, S is a union of equivalence classes under $\overset {K}{\cong }$ .

The dual Knuth equivalence (or dual plactic congruence) is an equivalence relation $\overset {K^*}{\cong }$ on $\mathfrak {S}_n$ defined by

$$\begin{align*}\sigma \overset{K^*}{\cong} \rho \quad \text{if and only if} \quad \sigma^{-1} \overset{K}{\cong} \rho^{-1}. \end{align*}$$

A subset S of $\mathfrak {S}_n$ is called dual plactic-closed if for any $\sigma \in S$ , every $\rho \in \mathfrak {S}_n$ with $\rho \overset {K^*}{\cong } \sigma $ is also an element of S; in other words, S is a union of equivalence classes under $\overset {K^*}{\cong }$ .

The Knuth and dual Knuth equivalences are closely related to the Robinson–Schensted correspondence, which is a one-to-one correspondence between $\mathfrak {S}_n$ and $\bigcup _{\lambda \vdash n} \mathrm {SYT}(\lambda ) \times \mathrm {SYT}(\lambda )$ . For $\sigma \in \mathfrak {S}_n$ , we use the notation $(\mathtt {ins}(\sigma ), \mathtt {rec}(\sigma ))$ to represent the image of $\sigma $ under this bijection. We call $\mathtt {ins}(\sigma )$ and $\mathtt {rec}(\sigma )$ as the insertion tableau and recording tableau of $\sigma $ , respectively. It is well known that $\mathtt {ins}(\sigma ) = \mathtt {rec}(\sigma ^{-1})$ and

$$ \begin{align*} \sigma \overset{K}{\cong} \rho \quad \text{if and only if} \quad \mathtt{ins}(\sigma) = \mathtt{ins}(\rho) \quad \text{for }\sigma, \rho \in \mathfrak{S}_n. \end{align*} $$

Putting these together, one can easily derive that

$$ \begin{align*} \sigma \overset{K^*}{\cong} \rho \quad \text{if and only if} \quad \mathtt{rec}(\sigma) = \mathtt{rec}(\rho) \quad \text{for }\sigma, \rho \in \mathfrak{S}_n. \end{align*} $$

For a subset S of $\mathfrak {S}_n$ , S is plactic-closed if and only if $\{\gamma ^{-1} \mid \gamma \in S\}$ is dual plactic-closed. Based on this fact, we will consider the claim that $\Sigma _R(P)$ is plactic-closed and the claim that $\Sigma _L(P)$ is dual plactic-closed to be identical.

Let us collect the terminologies and lemmas necessary for the proof of the main result of this subsection. Let T be a standard Young tableau of skew shape. Denote by $\mathrm {Rect}(T)$ the rectification of T – that is, the unique standard Young tableau of partition shape obtained by applying jeu de taquin slides to T (see [14, Section 1.2]). Then

(6.1) $$ \begin{align} \mathrm{Rect}(T) = \mathtt{ins}(\mathsf{read}_{\tau_0}(T) w_0) \quad \text{for any }T \in \mathrm{SYT}(\lambda/\mu). \end{align} $$

Define $T^{\mathrm {t}}$ to be the tableau obtained from T by flipping it along its main diagonal.

Lemma 6.1. Let $\lambda / \mu $ be a skew partition and $T \in \mathrm {SYT}(\lambda /\mu )$ . Then

$$\begin{align*}\mathtt{ins}(\mathsf{read}_\tau(T)) = \mathrm{Rect}(T)^{\mathrm{t}} \quad \text{for any }\tau \in \mathsf{DS}(\lambda/\mu). \end{align*}$$

Proof. It is well known that

(6.2) $$ \begin{align} \mathtt{ins}(\sigma w_0) = \mathtt{ins}(\sigma)^{\mathrm{t}} \quad \text{for any }\sigma \in \mathfrak{S}_n \end{align} $$

(for instance, see [31, Theorems 3.2.3]). Therefore, due to Equation (6.1), the assertion can be verified by showing that

(6.3) $$ \begin{align} \mathtt{ins}(\mathsf{read}_{\tau}(T)) = \mathtt{ins}(\mathsf{read}_{\tau^{\lambda/\mu}_0}(T)) \quad \text{ for any }\tau \in \mathsf{DS}(\lambda/\mu). \end{align} $$

Applying Taşkin’s result [36, Proposition 3.2.5] to the weak order³ on $\mathrm {SYT}_n$ given in [36, Definition 3.1.3], we derive that for $\sigma , \rho \in \mathfrak {S}_n$ with $\sigma \preceq _R \rho $ ,

(6.4) $$ \begin{align} \mathtt{ins}(\sigma) = \mathtt{ins}(\rho) \quad \text{or} \quad \mathrm{sh}(\mathtt{ins}(\rho)) \triangleleft \mathrm{sh}(\mathtt{ins}(\sigma)). \end{align} $$

Here, $\trianglelefteq $ denotes the dominance order on the set of partitions of n. And, Lemma 3.7 says that

(6.5) $$ \begin{align} \mathsf{read}_{\tau_0^{\lambda/\mu}}(T) \preceq_R \mathsf{read}_{\tau}(T) \preceq_R \mathsf{read}_{\tau_1^{\lambda/\mu}}(T) \quad \text{for }\tau \in \mathsf{DS}(\lambda/\mu). \end{align} $$

Note that

$$\begin{align*}\mathtt{ins}(\mathsf{read}_{\tau^{\lambda^{\mathrm{t}}/\mu^{\mathrm{t}}}_0}(T^{\mathrm{t}})w_0) \underset{Eq. ({6.1})}{=} \mathrm{Rect}(T^{\mathrm{t}}) = \mathrm{Rect}(T)^{\mathrm{t}} \underset{Eq. ({6.1})}{=} \mathtt{ins}(\mathsf{read}_{\tau^{\lambda/\mu}_0}(T)w_0)^{\mathrm{t}}. \end{align*}$$

Since $\mathsf {read}_{\tau ^{\lambda /\mu }_1}(T) = \mathsf {read}_{\tau ^{\lambda ^{\mathrm {t}}/\mu ^{\mathrm {t}}}_0}(T^{\mathrm {t}}) w_0$ , it follows from Equation (6.2) that

(6.6) $$ \begin{align} \mathtt{ins}(\mathsf{read}_{\tau^{\lambda/\mu}_1}(T)) = \mathtt{ins}(\mathsf{read}_{\tau^{\lambda/\mu}_0}(T)). \end{align} $$

Now, the equality in Equation (6.3) is obtained by combining Equation (6.4), Equation (6.5) and Equation (6.6).

We introduce two important results due to Malvenuto [27].

Lemma 6.2 [27, Theorem 1]

For $P \in \mathsf {P}_n$ , if $\Sigma _R(P)$ is plactic-closed, then P is a Schur labeled skew shape poset.

For $P \in \mathsf {P}_n$ , we say a subposet Q of P is convex if Q satisfies the property that for any $x \in P$ if there exist $y_1, y_2 \in Q$ such that $y_1 \preceq _P x \preceq _P y_2$ , then $x \in Q$ . For a subposet $Q = \{i_1 < i_2 < \cdots <i_{|Q|} \}$ of $P \in \mathsf {P}_n$ , the standardization of Q, denoted by $\mathsf {st}(Q)$ , is the poset obtained from Q by replacing $i_j$ with j for $1 \leq j \leq |Q|$ .

Lemma 6.3 [27, Corollary 1]

Let $P \in \mathsf {P}_n$ such that $\Sigma _R(P)$ is plactic-closed. For any convex subposet Q of P, $\Sigma _R(\mathsf {st}(Q))$ is plactic-closed.

Now, we are ready to prove the main result of this subsection.

Theorem 6.4. For $P \in \mathsf {P}_n$ , P is a regular Schur labeled skew shape poset if and only if $\Sigma _L(P)$ is dual plactic-closed.

Proof. To establish the ‘only if’ part, let $P \in \mathsf {RSP}_n$ and $\lambda /\mu = \mathrm {sh}(\tau _P)$ . Due to Lemma 3.2, we have that

$$\begin{align*}\Sigma_L(P) = \mathsf{read}_{\tau_P}(\mathrm{SYT}(\lambda/\mu)). \end{align*}$$

We claim that $\mathsf {read}_{\tau _P}(\mathrm {SYT}(\lambda /\mu ))$ is dual plactic-closed.

As mentioned in [15, Property A], one can easily see that $\mathsf {read}_{\tau _0}(\mathrm {SYT}(\lambda /\mu ))$ is dual plactic-closed.⁴ In addition, by Lemma 6.1, we have

(6.7) $$ \begin{align} \mathtt{ins}(\mathsf{read}_{\tau_P}(T)) = \mathtt{ins}(\mathsf{read}_{\tau_0}(T)) \quad \text{for all }T \in \mathrm{SYT}(\lambda/\mu). \end{align} $$

Therefore, given $T \in \mathrm {SYT}(\lambda /\mu )$ , if we show that

$$\begin{align*}\mathsf{read}_{\tau_P}(T) \overset{K^*}{\cong} \mathsf{read}_{\tau_P}(U) \quad \text{ for all }U \in \mathrm{SYT}(\lambda/\mu)\text{ with }\mathsf{read}_{\tau_0}(T) \overset{K^*}{\cong} \mathsf{read}_{\tau_0}(U), \end{align*}$$

then we have

$$\begin{align*}\{\gamma \in \mathfrak{S}_n \mid \gamma \overset{K^*}{\cong} \mathsf{read}_{\tau_P}(T)\} \subseteq \mathsf{read}_{\tau_P}(\mathrm{SYT}(\lambda/\mu)). \end{align*}$$

Let $T, U \in \mathrm {SYT}(\lambda /\mu )$ with $\mathsf {read}_{\tau _0}(T) \overset {K^*}{\cong } \mathsf {read}_{\tau _0}(U)$ . Since $\mathsf {read}_{\tau _0}(\mathrm {SYT}(\lambda /\mu ))$ is dual plactic-closed, there exist standard Young tableaux $T_0 := T, T_1, \ldots , T_l := U$ of shape $\lambda /\mu $ such that for any $1 \leq k \leq l$ ,

$$ \begin{align*} \mathsf{read}_{\tau_0}(T_k) \overset{K^*}{\cong}\mathsf{read}_{\tau_0}(T) \ \ \text{and} \ \ \mathsf{read}_{\tau_0}(T_k) = s_{i_k} \mathsf{read}_{\tau_0}(T_{k-1}) \ \ \text{for some }i_k \in [n-1]. \end{align*} $$

Combining Equation (6.7) with the equality $\mathrm {sh}(\mathtt {ins}(\mathsf {read}_{\tau _0}(T_k))) = \mathrm {sh}(\mathtt {ins}(\mathsf {read}_{\tau _0}(T)))$ , we have

(6.8) $$ \begin{align} \mathrm{sh}(\mathtt{ins}(\mathsf{read}_{\tau_P}(T_k))) = \mathrm{sh}(\mathtt{ins}(\mathsf{read}_{\tau_P}(T))) \quad \text{for all }1 \le k \le l. \end{align} $$

Note that Equation (6.4) is equivalent to the statement that for $\sigma , \rho \in \mathfrak {S}_n$ with $\sigma \preceq _L \rho $ ,

(6.9) $$ \begin{align} \sigma \overset{K^*}{\cong} \rho \quad \text{or} \quad \mathrm{sh}(\mathtt{rec}(\rho)) \triangleleft \mathrm{sh}(\mathtt{rec}(\sigma)). \end{align} $$

Putting Equation (6.8) together with Equation (6.9), we have $\mathsf {read}_{\tau _P}(T) \overset {K^*}{\cong } \mathsf {read}_{\tau _P}(U)$ . Since we chose arbitrary $T,U \in \mathrm {SYT}(\lambda /\mu )$ , we conclude that $\mathsf {read}_{\tau _P}(\mathrm {SYT}(\lambda /\mu ))$ is dual plactic-closed.

To establish the ‘if’ part of the assertion, we prove the contraposition; that is, if P is not a regular Schur labeled skew shape poset, then $\Sigma _L(P)$ is not dual plactic-closed. If P is not a Schur labeled skew shape poset, then Lemma 6.2 says that $\Sigma _L(P)$ is not dual plactic-closed. So, we assume that $P \in \mathsf {P}_n$ is a non-regular Schur labeled skew shape poset.

One can easily check that if $n = 1,2,3$ , then $\Sigma _L(P)$ is not dual plactic-closed. Suppose $n> 3$ . Then, by Lemma 3.6, $\tau _P$ is a non-distinguished Schur labeling. This implies that there exists $k \in \mathbb Z_{>0}$ such that $\mathsf {cnt}_{k}(\tau _P)$ is not filled with consecutive integers. Let $k_0$ be the minimum among these integers and let $m_0$ be the minimum element among $m \in \mathsf {cnt}_{k_0}(\tau _P)$ such that $m + 1 \notin \mathsf {cnt}_{k_0}(\tau _P)$ . Since $\mathsf {cnt}_{k_0}(\tau _P)$ is not filled with consecutive integers, we can choose

$$ \begin{align*}m_1 = \min \{m \in \mathsf{cnt}_{k_0}(\tau_P) \mid m> m_0\}. \end{align*} $$

Since $m_0$ and $m_1$ are in the same connected component of the Schur labeling $\tau _P$ and $m_0 < m_1$ , we can take $m_{-1} \in \mathsf {cnt}_{k_0}(\tau _P)$ such that $m_{-1} < m_1$ and $m_{-1}$ is adjacent to $m_1$ . Here, the sentence ‘ $m_{-1}$ is adjacent to $m_1$ ’ means that the box containing $m_{-1}$ and that containing $m_{1}$ share an edge. We note that $m_{-1}$ can be $m_0$ . Because of the choice of $m_1$ , we have $m_{-1} < m_0+1 < m_1$ . Let Q be the subposet of P whose underlying set is $\{m_{-1}, m_0 + 1, m_1\}$ . In P, $m_1$ covers $m_{-1}$ and $m_0 + 1$ is incomparable with both $m_1$ and $m_{-1}$ . This implies that Q is a convex subposet of P. In addition, since $m_{-1} < m_0 + 1 < m_1$ , we have $\Sigma _L(\mathsf {st}(Q)) = \{123,132,213\}$ or $\Sigma _L(\mathsf {st}(Q)) = \{312,231,321\}$ . Thus, $\Sigma _L(\mathsf {st}(Q))$ is not dual plactic closed. Combining this with Lemma 6.3 yields that $\Sigma _L(P)$ is not dual plactic closed, as desired.

6.2 Distinguished filtrations of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$

We begin by introducing the definition of distinguished filtrations.

Definition 6.5. Let $\mathcal {B} = \{\mathcal {B}_\alpha \mid \alpha \in I\}$ be a linearly independent subset of $\mathrm {QSym}_n$ with the property that $\mathcal {B}_\alpha $ is F-positive for all $\alpha \in I$ , where I is an index set. Given a finite dimensional $H_n(0)$ -module M, a distinguished filtration of M with respect to $\mathcal {B}$ is an $H_n(0)$ -submodule series of M

$$\begin{align*}0 =: M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_l := M \end{align*}$$

such that for all $1 \leq k \leq l$ , $\mathrm {ch}([M_k / M_{k-1}]) = \mathcal {B}_\alpha $ for some $\alpha \in I$ .

As seen in Example 6.6, a distinguished filtration of M with respect to $\mathcal {B}$ may not exist even if $\mathrm {ch}([M])$ expands positively in $\mathcal {B}$ . This is because the category $H_n(0)\text{-}\mathbf {mod}$ is neither semisimple nor representation-finite when $n> 3$ ([10, 11]).

Example 6.6. Let $\mathcal {B} = \{s_\lambda \mid \lambda \vdash 4 \}$ . For $B = \{2314, 1423, 3214,2413, 1432, 3412\}$ , let M be the $H_4(0)$ -module with underlying space $\mathbb C B$ and with the $H_4(0)$ -action defined by

$$ \begin{align*} \pi_{i} \cdot \gamma := \begin{cases} \gamma & \text{if }i \in \mathrm{Des}_L(\gamma), \\ 0 & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \notin B, \\ s_i \gamma & \text{if }i \notin \mathrm{Des}_L(\gamma)\text{ and }s_i\gamma \in B. \end{cases} \end{align*} $$

The $H_4(0)$ -action on $B \cup \{ 0 \}$ is illustrated in the following figure:

One sees that

$$\begin{align*}\mathrm{ch}([M]) = s_{(3,1)} + s_{(2,1,1)} = (F_{(3,1)} + F_{(2,2)} + F_{(1,3)}) + (F_{(2,1,1)} + F_{(1,2,1)} + F_{(1,1,2)}). \end{align*}$$

So, if there exists a distinguished filtration of M with respect to $\mathcal {B}$ , then there exists a three-dimensional $H_4(0)$ -submodule N of M such that $\mathrm {ch}([N])$ is equal to either $s_{(3,1)}$ or $s_{(2,1,1)}$ . We claim that such a submodule N does not exist.

Note that

(6.10) $$ \begin{align} M = \mathbb C \{2314 - 3214, 1423-1432, 2413, 3412 \} \oplus \mathbb C \{3214\} \oplus \mathbb C \{1432\}. \end{align} $$

Here, $\mathbb C \{3214\}$ and $\mathbb C \{1432\}$ are irreducible. And $\mathbb C \{2314 - 3214, 1423-1432, 2413, 3412 \}$ is indecomposable since it is isomorphic to a submodule of the injective indecomposable module $\mathbf {P}_{(1,2,1)}$ . Therefore, Equation (6.10) is a decomposition of M into indecomposables. The $H_4(0)$ -action on $\{2314 - 3214, 1423-1432, 2413, 3412, 3214, 1432\} \cup \{ 0 \}$ is illustrated in the following figure:

The injective hulls of $\mathbb C \{2314 - 3214, 1423-1432, 2413, 3412 \}$ , $\mathbb C \{3214\}$ and $\mathbb C \{1432\}$ are $\mathbf {P}_{(1,2,1)}$ , $\mathbf {P}_{(1,3)}$ and $\mathbf {P}_{(3,1)}$ , respectively. This implies that the socle of M is $\mathbb C\{3412\} \oplus \mathbb C \{3214\} \oplus \mathbb C \{1432\}$ . It follows that for every three-dimensional submodule N of M, $1 \le \dim \mathrm {soc}(N) \le 3$ . We list all three-dimensional submodules N of M in Table 1. Based on this, we conclude that there are no $H_4(0)$ -submodules N of M such that $\mathrm {ch}([N]) = s_{(3,1)}$ or $s_{(2,1,1)}$ .

Table 1 The complete list of three-dimensional submodules of M in Example 6.6.

Let $f \in \mathrm {QSym}_n$ and $\mathcal {B} = \{\mathcal {B}_\alpha \mid \alpha \in I \}$ be the linearly independent set given in Definition 6.5. When f expands positively in $\mathcal {B}$ , that is,

(6.11) $$ \begin{align} f = \sum_{\alpha \in I} c_\alpha \mathcal{B}_\alpha \quad (c_\alpha \in \mathbb Z_{\geq 0}), \end{align} $$

finding an $H_n(0)$ -module M such that

1. (C1) $\mathrm {ch}([M]) = f$ ,

2. (C2) it is not a direct sum of irreducible modules, yet it possesses a combinatorial model that can be effectively handled, and

3. (C3) it has a distinguished filtration with respect to $\mathcal {B}$

is a very important problem in the sense that this filtration can be considered as a nice representation theoretic interpretation of Equation (6.11).

In this subsection, we focus on the above problem in the case where $\mathcal {B}$ is $\mathcal {S} := \{s_{\lambda } \mid \lambda \vdash n \}$ and $f = s_{\lambda /\mu }$ for a skew partition $\lambda /\mu $ of size n. Note that for all $P \in \mathsf {RSP}_n$ with $\mathrm {sh}(\tau _P) = \lambda /\mu $ , $\mathsf {M}_P$ satisfies (C1) and (C2) because $\mathrm {ch}([\mathsf {M}_P]) = s_{\lambda /\mu }$ by Theorem 2.9(2) and it has a combinatorial model $\Sigma _L(P)$ . In the following, we show that $\mathsf {M}_P$ satisfies (C3).

Theorem 6.7. For every $P \in \mathsf {RSP}_n$ , $\mathsf {M}_P$ has a distinguished filtration with respect to $\mathcal {S}$ .

Proof. To begin with, we choose any total order $\ll $ on $\mathrm {SYT}_n$ subject to the condition that

(6.12) $$ \begin{align} T \ll S \quad \text{whenever }\mathrm{sh}(T) \triangleleft \mathrm{sh}(S). \end{align} $$

Write $\{\mathtt {rec}(\gamma ) \mid \gamma \in \Sigma _L(P) \}$ as

$$\begin{align*}\{T_1 \ll T_2 \ll \cdots \ll T_l \}. \end{align*}$$

For $0 \leq k \leq l$ , set

$$\begin{align*}B_k := \{\gamma \in \mathfrak{S}_n \mid \mathtt{rec}(\gamma) = T_i \ \text{for some }1 \leq i \leq k\}. \end{align*}$$

It is clear that $\emptyset = B_0 \subset B_1 \subset B_2 \subset \cdots \subset B_l$ . And, by Theorem 6.4, we have $B_l = \Sigma _L(P)$ . We claim that

(6.13) $$ \begin{align} 0 = \mathbb C B_0 \subset \mathbb C B_1 \subset \mathbb C B_2 \subset \cdots \subset \mathbb C B_l = \mathsf{M}_{P} \end{align} $$

is a distinguished filtration of $\mathsf {M}_P$ with respect to $\mathcal {S}$ .

First, we show that for $1 \leq k \leq l$ ,

$$\begin{align*}\pi_i \cdot \gamma \in B_k \cup \{0\} \quad \text{for all }i \in [n-1]\text{ and }\gamma \in B_k. \end{align*}$$

Take any $i \in [n-1]$ and $\gamma \in B_k$ . If $\pi _i \cdot \gamma = 0$ or $\gamma $ , then there is nothing to prove. Assume that $\pi _i \cdot \gamma = s_i \gamma $ . Then, by the definition of $H_n(0)$ -action on $\mathsf {M}_P$ , we have $\gamma \preceq _L s_i \gamma $ . Combining this inequality with Equation (6.9) yields that

$$\begin{align*}\gamma \overset{K^*}{\cong} s_i \gamma \quad \text{or} \quad \mathrm{sh}(\mathtt{rec}(s_i \gamma)) \triangleleft \mathrm{sh}(\mathtt{rec}(\gamma)). \end{align*}$$

This implies that $s_i \gamma \in B_k$ , as desired.

Next, we show that the filtration given in Equation (6.13) is distinguished with respect to $\mathcal {S}$ . For $1 \le k \le l$ , $\{\gamma + M_{k-1} \mid \gamma \in B_k \setminus B_{k-1}\}$ is a basis for $M_k / M_{k-1}$ and $B_k \setminus B_{k-1}$ is an equivalence class under $\overset {K^*}{\cong }$ . It follows that $\mathrm {ch}([M_k / M_{k-1}])$ is a Schur function; more precisely, $\mathrm {ch}([M_k / M_{k-1}]) = s_{\mathrm {sh}(T_k)^{\mathrm {t}}}$ .

Example 6.8. Let $P = \mathsf {poset}(\tau _0^{(4,2,1)/(2,1)})$ . Following the method presented in the proof of Theorem 6.7, we will construct two distinguished filtrations of $\mathsf {M}_P$ with respect to $\{s_{\lambda } \mid \lambda \vdash 4 \}$ by choosing two distinct total orders on $\mathrm {SYT}_4$ .

Note that $\{\mathtt {rec}(\gamma ) \mid \gamma \in \Sigma _L(P)\}$ is given by

Figure 2 The $H_4(0)$ -action on the basis $\Sigma _L(P) = [2134, 4321]_L$ for $\mathsf {M}_P$ and the sets $B^{\prime}_k (1 \leq k \leq 5)$ in Example 6.8.

and $\mathrm {sh}(Q_1) \triangleleft \mathrm {sh}(Q_2) = \mathrm {sh}(Q_3) \triangleleft \mathrm {sh}(Q_4) \triangleleft \mathrm {sh}(Q_5)$ . Choose a total order $\ll _1$ (resp. $\ll _2$ ) on $\mathrm {SYT}_4$ satisfying both Equation (6.12) and $Q_2 \ll _1 Q_3$ (resp. $Q_3 \ll _2 Q_2$ ). For $1 \leq k \leq 5$ , let

$$\begin{align*}B^{\prime}_k := \{\gamma \in \mathfrak{S}_4 \mid \mathtt{rec}(\gamma) = Q_k\}. \end{align*}$$

When we use $\ll _1$ , we let

$$ \begin{align*}B_k := \bigsqcup_{1 \le l \le k} B^{\prime}_l \quad \text{for }0 \le k \le 5, \end{align*} $$

and when we use $\ll _2$ , we let

$$ \begin{align*}B_k := \bigsqcup_{1 \le l \le k} B^{\prime}_l \quad \text{for }k = 0,1,3,4,5 \quad \text{and} \quad B_2 := B^{\prime}_1 \sqcup B^{\prime}_3. \end{align*} $$

Then

$$\begin{align*}0 = \mathbb C B_0 \subset \mathbb C B_1 \subset \mathbb C B_2 \subset \mathbb C B_3 \subset \mathbb C B_4 \subset \mathbb C B_5 = \mathsf{M}_P \end{align*}$$

is the desired distinguished filtration of $\mathsf {M}_P$ with respect to $\{s_\lambda \mid \lambda \vdash 4 \}$ .

For the readers’ convenience, we draw the $H_4(0)$ -action on the basis $\Sigma _L(P) = [2134, 4321]_L$ for $\mathsf {M}_P$ and the sets $B^{\prime}_k (1 \leq k \leq 5)$ in Figure 2.

7 Further avenues

In this section, we discuss future directions regarding the classification problem, the decomposition problem, and how to recover $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ from a module of the generic Hecke algebra $H_n(q)$ by specializing q to $0$ .

7.1 The classification problem

In Theorem 5.5, we successfully classify $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ . To be precise, we show that for $P, Q \in \mathsf {RSP}_n$ ,

(7.1) $$ \begin{align} \mathsf{M}_P \cong \mathsf{M}_Q \quad \text{if and only if} \quad \mathrm{sh}(\tau_P) = \mathrm{sh}(\tau_Q). \end{align} $$

Recall that $\mathsf {RSP}_n = \mathsf {RP}_n \cap \mathsf {SP}_n$ . Hence, it would be natural to consider the classification problem for $\{\mathsf {M}_P \mid P \in \mathsf {SP}_n\}$ and $\{\mathsf {M}_P \mid P \in \mathsf {RP}_n\}$ .

7.1.1 A remark on the classification problem for $\{\mathsf {M}_P \mid P \in \mathsf {SP}_n \}$

Since the notion ‘the shape of $\tau _P$ ’ is available for $P \in \mathsf {SP}_n$ , one may expect that the classification given in Equation (7.1) can be extended to $\{\mathsf {M}_P \mid P \in \mathsf {SP}_n\}$ . Unfortunately, this expectation turns out to be false.

Let

Then $\mathsf {M}_{\mathsf {poset}(\tau _i)} (i = 1,2,3)$ is decomposed into indecomposables as follows:

$$ \begin{align*} \mathsf{M}_{\mathsf{poset}(\tau_1)} & \cong \mathbf{P}_{(4)} \oplus \mathbf{P}_{(2,2)}, \\ \mathsf{M}_{\mathsf{poset}(\tau_2)} & \cong \mathbf{F}_{(1,2,1)} \oplus \mathsf{B}(4213, 4312) \oplus \mathbf{F}_{(3,1)} \oplus \mathbf{F}_{(2,2)} \oplus \mathbf{F}_{(4)}, \\ \mathsf{M}_{\mathsf{poset}(\tau_3)} & \cong \mathbf{F}_{(1,2,1)} \oplus \mathsf{B}(4213, 4312) \oplus \mathsf{B}(2431, 3421) \oplus \mathbf{F}_{(4)}, \end{align*} $$

where $\mathsf {B}(4213, 4312)$ and $\mathsf {B}(2431, 3421)$ are $2$ -dimensional indecomposable modules. These decompositions show that $\mathsf {M}_{\mathsf {poset}(\tau _1)}$ , $\mathsf {M}_{\mathsf {poset}(\tau _2)}$ and $\mathsf {M}_{\mathsf {poset}(\tau _3)}$ are pairwise non-isomorphic although all $\tau _{\mathsf {poset}(\tau _i)}$ ’s have the same shape.

Table 2 Seven pairs $(I_1^{(k)}, I_2^{(k)})$ in $\mathfrak {A}_6$ .

7.1.2 A conjecture on the classification problem for $\{\mathsf {M}_P \mid P \in \mathsf {RP}_n \}$

Note that for $P \in \mathsf {RP}_n$ , the notion ‘the shape of $\tau _P$ ’ has not been defined. This leads us to introduce a classification of $\{\mathsf {M}_P \mid P \in \mathsf {RSP}_n\}$ without using this notion. To be precise, by combining Theorem 4.7 and Theorem 5.5, we derive that for $P, Q \in \mathsf {RSP}_n$ ,

(7.2) $$ \begin{align} \mathsf{M}_P \cong \mathsf{M}_Q \quad \text{if and only if} \quad \Sigma_L(P) \overset{D}{\simeq} \Sigma_L(Q). \end{align} $$

We expect that this classification can be extended to $\mathsf {RP}_n$ in its current form. The validity of this expectation has been checked for values of n up to 6 with the aid of the computer program SageMath. Let us provide an overview of our verification process. We first classify all left weak Bruhat intervals in $\mathfrak {S}_n$ ( $n \leq 6$ ) up to descent-preserving isomorphism and choose a complete list $\mathfrak {I}_n$ of inequivalent representatives. Next, we let $\mathfrak {A}_n$ be the set of all unordered pairs $([\sigma _1, \rho _1]_L, [\sigma _2, \rho _2]_L)$ of intervals in $\mathfrak {I}_n$ satisfying that $[\sigma _1, \rho _1]_L \neq [\sigma _2, \rho _2]_L$ and

(7.3) $$ \begin{align} \mathrm{ch}([\mathsf{B}(\sigma_1, \rho_1)]) = \mathrm{ch}([\mathsf{B}(\sigma_2, \rho_2)]), \;\, \mathrm{Des}_L(\sigma_1) = \mathrm{Des}_L(\sigma_2), \;\, \mathrm{Des}_L(\rho_1) = \mathrm{Des}_L(\rho_2). \end{align} $$

Note that Equation (7.3) is a necessary condition for $\mathsf {B}(\sigma _1, \rho _1) \cong \mathsf {B}(\sigma _2, \rho _2)$ . Finally, we show that for all $(I_1, I_2) \in \mathfrak {A}_n$ , $\mathsf {B}(I_1) \not \cong \mathsf {B}(I_2)$ . When $n \leq 5$ , there is nothing to prove because $\mathfrak {A}_n = \emptyset $ . When $n = 6$ , $\mathfrak {A}_6$ has fourteen pairs. Note that if $(I_1, I_2) \in \mathfrak {A}_6$ , then $(w_0 \cdot I_1 \cdot {w_0}, w_0 \cdot I_2 \cdot {w_0}) \in \mathfrak {A}_6$ and

$$\begin{align*}\mathsf{B}(I_1) \cong \mathsf{B}(I_2) \quad \underset{\text{[{19}, Theorem 4]}}{\Longleftrightarrow} \quad \mathsf{B}(w_0 \cdot I_1 \cdot w_0) \cong \mathsf{B}(w_0 \cdot I_2 \cdot w_0). \end{align*}$$

Therefore, it suffices to examine seven pairs $(I_1^{(k)}, I_2^{(k)})$ listed in Table 2. For $3 \leq k \leq 7$ , using Lemma 5.2, one can see that the projective covers of $\mathsf {B}(I_1^{(k)})$ and $\mathsf {B}(I_2^{(k)})$ are not isomorphic. Therefore, $\mathsf {B}(I_1^{(k)})$ and $\mathsf {B}(I_2^{(k)})$ are not isomorphic. For $k = 1, 2$ , one can see that $\mathsf {B}(I_1^{(k)})$ and $\mathsf {B}(I_2^{(k)})$ are not isomorphic in a brute force manner.

Let us give another evidence for our expectation. Specifically, we show that Equation (7.2) holds when $P \in \mathsf {RSP}_n$ , $Q \in \mathsf {RP}_n$ , and $\mathrm {ch}([\mathsf {M}_P])$ is a Schur function. This can be derived from the proposition presented below.

Proposition 7.1. Let P be a poset in $\mathsf {RSP}_n$ such that $\mathrm {ch}([\mathsf {M}_P])$ is a Schur function.

1. (1) If $Q \in \mathsf {P}_n$ satisfies that $\mathsf {M}_Q \cong \mathsf {M}_P$ , then $Q \in \mathsf {RSP}_n$ .

2. (2) The isomorphism class of $\mathsf {M}_P$ within $\{\mathsf {M}_Q \mid Q \in \mathsf {P}_n \}$ is equal to the isomorphism class of $\mathsf {M}_P$ within $\{\mathsf {M}_Q \mid Q \in \mathsf {RSP}_n \}$ as sets.

Proof. (1) Suppose that $\mathrm {ch}([\mathsf {M}_P]) = s_{\lambda }$ for some $\lambda \vdash n$ . By [39, Theorem 2.2], $\mathrm {sh}(\tau _P)$ is either $\lambda $ or $\lambda ^\circ $ , where $\lambda ^\circ $ denotes the skew partition whose Young diagram is obtained by rotating $\mathtt {yd}(\lambda )$ by $180^\circ $ .

First, we consider the case where $\mathrm {sh}(\tau _P) = \lambda $ . Let $f: \mathsf {M}_P \rightarrow \mathsf {M}_Q$ be an $H_n(0)$ -module isomorphism. By Theorem 3.9, we see that $\Sigma _L(P) = [\mathsf {read}_{\tau _P} (T_{\lambda }), \mathsf {read}_{\tau _P}(T^{\prime}_{\lambda })]_L$ , and therefore, $\mathsf {read}_{\tau _P} (T_{\lambda })$ is a cyclic generator of $\mathsf {M}_P$ . In addition, in view of [32, Lemma 3.12], we have that

(7.4) $$ \begin{align} \mathrm{Des}_L(\mathsf{read}_{\tau_P}(T)) \not\supseteq \mathrm{Des}_L(\mathsf{read}_{\tau_P}(T_{\lambda})) \text{ for all }T \in \mathrm{SYT}(\lambda) \setminus \{T_{\lambda} \}. \end{align} $$

Combining (7.4) with the equality $\mathrm {ch}([\mathsf {M}_P]) = \mathrm {ch}([\mathsf {M}_Q])$ , we can deduce that there exists a unique $\sigma \in \Sigma _L(Q)$ such that $\mathrm {Des}_L(\sigma ) \supseteq \mathrm {Des}_L(\mathsf {read}_{\tau _P}(T_{\lambda }))$ . This fact implies that $f(\mathsf {read}_{\tau _P}(T_{\lambda })) = c \sigma $ for some nonzero $c \in \mathbb C$ . We may assume that $c = 1$ by considering the isomorphism $\frac {1}{c}f$ instead of f. Since f is an $H_n(0)$ -module isomorphism, $\Sigma _L(Q)$ is equal to $f(\Sigma _L(P))$ and therefore is a left weak Bruhat interval. Furthermore, it holds that

$$\begin{align*}\mathrm{Des}_L(f(\gamma)) = \mathrm{Des}_L(\gamma) \quad \text{for all }\gamma \in \Sigma_L(P). \end{align*}$$

As a consequence, we obtain a descent-preserving isomorphism $f|_{\Sigma _L(P)}: \Sigma _L(P) \rightarrow \Sigma _L(Q)$ . Now the assertion follows from Theorem 4.7.

Next, consider the case where $\mathrm {sh}(\tau _P) = \lambda ^\circ $ . Let $\overline {P}^*$ and $\overline {Q}^*$ be the posets in $\mathsf {P}_n$ whose orders are defined by

$$\begin{align*}u \preceq_{\overline{P}^*} v \Longleftrightarrow n + 1 - v \preceq_P n + 1 - u \quad \text{and} \quad u \preceq_{\overline{Q}^*} v \Longleftrightarrow n + 1 - v \preceq_Q n + 1 - u, \end{align*}$$

respectively. Since P is a poset in $\mathsf {RSP}_n$ with $\mathrm {sh}(\tau _P) = \lambda ^\circ $ , $\overline {P}^*$ is a poset in $\mathsf {RSP}_n$ with $\mathrm {sh}(\tau _{\overline {P}^*}) = \lambda $ . By [9, Theorem 3.6(a)], we have $\mathsf {M}_{\overline {P}^*} \cong \mathbf {T}^+_{\unicode{x3c6} } (\mathsf {M}_{P})$ and $\mathsf {M}_{\overline {Q}^*} \cong \mathbf {T}^+_{\unicode{x3c6} } (\mathsf {M}_{Q})$ , which implies that $\mathsf {M}_{\overline {Q}^*} \cong \mathsf {M}_{\overline {P}^*}$ . It follows from the first case that $\overline {Q}^* \in \mathsf {RSP}_n$ , thus $Q \in \mathsf {RSP}_n$ .

(2) It follows from (1).

Based on these evidences, we propose the following conjecture.

Conjecture 7.2. Let $P, Q \in \mathsf {RP}_n$ . If $\mathsf {M}_P \cong \mathsf {M}_Q$ , then $\Sigma _L(P) \overset {D}{\simeq } \Sigma _L(Q)$ .

We remark that the converse of Conjecture 7.2 holds due to Proposition 4.1.

7.2 The decomposition problem of $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$

A Young diagram of skew shape is called a ribbon if it does not contain any $2\times 2$ square. For simplicity, we call a skew partition a ribbon if the corresponding Young diagram is a ribbon. Note that our ribbons are not necessarily connected. Consider a skew partition

$$\begin{align*}\lambda/\mu = \lambda^{(1)} / \mu^{(1)} \star \lambda^{(2)} / \mu^{(2)} \star \cdots \star \lambda^{(k)} / \mu^{(k)} \end{align*}$$

such that $\lambda ^{(i)} / \mu ^{(i)}$ is connected for all $1 \le i \le k$ . We say that $\lambda / \mu $ contains a disconnected ribbon if there exists an index $1 \le j \le k-1$ such that both $\lambda ^{(j)} / \mu ^{(j)}$ and $\lambda ^{(j+1)} / \mu ^{(j+1)}$ are ribbons. With this notation, we state the following proposition.

Proposition 7.3. Let $P \in \mathsf {RSP}_n$ .

1. (1) If $\mathrm {sh}(\tau _P)$ is connected, then $\mathsf {M}_P$ is indecomposable.

2. (2) If $\mathrm {sh}(\tau _P)$ contains a disconnected ribbon, then $\mathsf {M}_P$ is not indecomposable.

Proof. (1) It follows from Lemma 5.4.

(2) Suppose that $\mathrm {sh}(\tau _P)$ contains a disconnected ribbon. Let $\lambda /\mu = \mathrm {sh}(\tau _P)$ . Write $\lambda /\mu $ as $\lambda ^{(1)} / \mu ^{(1)} \star \lambda ^{(2)} / \mu ^{(2)} \star \cdots \star \lambda ^{(k)} / \mu ^{(k)}$ , where $\lambda ^{(i)} / \mu ^{(i)}$ is connected for all $1 \le i \le k$ and both $\lambda ^{(j)} / \mu ^{(j)}$ and $\lambda ^{(j+1)} / \mu ^{(j+1)}$ are ribbons for some $1 \leq j \leq k-1$ .

In Appendix A, we constructed an $H_n(0)$ -module $X_{\lambda /\mu }$ satisfying that $X_{\lambda /\mu } \cong \mathsf {M}_P$ . From now on, we will prove the assertion for $X_{\lambda /\mu }$ instead of $\mathsf {M}_P$ . By Proposition A.2(1), we have the $H_n(0)$ -module isomorphism

$$\begin{align*}X_{\lambda/\mu} \cong X_{\lambda^{(1)}/\mu^{(1)}} \boxtimes \cdots \boxtimes X_{\lambda^{(k)}/\mu^{(k)}}. \end{align*}$$

Set $X^{(1)} := X_{\lambda ^{(1)} / \mu ^{(1)}} \boxtimes \cdots \boxtimes X_{\lambda ^{(j-1)} / \mu ^{(j-1)}}$ and $X^{(2)} := X_{\lambda ^{(j+2)} / \mu ^{(j+2)}} \boxtimes \cdots \boxtimes X_{\lambda ^{(k)} / \mu ^{(k)}}$ . Since $\lambda ^{(j)}/\mu ^{(j)}$ and $\lambda ^{(j+1)}/\mu ^{(j+1)}$ are ribbons, $X_{\lambda ^{(j)}/\mu ^{(j)}} \cong \mathbf {P}_\alpha $ and $X_{\lambda ^{(j+1)}/\mu ^{(j+1)}} \cong \mathbf {P}_\beta $ , where $\alpha = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{(j)} / \mu ^{(j)})$ and $\beta = {\boldsymbol {\unicode{x3b1} }}_{\mathrm {proj}}(\lambda ^{(j+1)} / \mu ^{(j+1)})$ . Therefore,

$$\begin{align*}X_{\lambda/\mu} \cong X^{(1)} \boxtimes \mathbf{P}_\alpha \boxtimes \mathbf{P}_\beta \boxtimes X^{(2)}. \end{align*}$$

Combining Lemma 5.1 with the fact that $\boxtimes $ is distributive over $\oplus $ , we derive the $H_n(0)$ -module isomorphism

$$\begin{align*}X_{\lambda/\mu} \cong (X^{(1)} \boxtimes \mathbf{P}_{\alpha \cdot \beta} \boxtimes X^{(2)}) \oplus (X^{(1)} \boxtimes \mathbf{P}_{\alpha \odot \beta} \boxtimes X^{(2)}). \end{align*}$$

This shows $X_{\lambda /\mu }$ is not indecomposable.

The contraposition of Proposition 7.3(2) says that if $\mathsf {M}_P$ is indecomposable, then $\mathrm {sh}(\tau _P)$ does not contain any disconnected ribbon. We ask if the converse is true. In the case where $\mathrm {sh}(\tau _P)$ is connected, it is true by Proposition 7.3(1). In the case where $\mathrm {sh}(\tau _P)$ is disconnected, we verified its validity when $|P| \leq 6$ . Indeed, this was done by showing that $\mathrm {End}(\mathsf {M}_P)$ has no idempotent except for $0$ and $\mathrm {id}$ . Refer to the following example.

Example 7.4. Let $\lambda /\mu = (3,3,1)/(1,1)$ and $P = \mathsf {poset}(\tau _0^{\lambda /\mu })$ . Then, $\Sigma _L(P) = [21435, 42531]_L$ is a basis for $\mathsf {M}_P$ . Let $f \in \mathrm {End}(\mathsf {M}_P)$ be an idempotent and let

$$ \begin{align*}f(21435) = \sum_{\gamma \in [21435, 42531]_L} c_\gamma \gamma \quad (c_\gamma \in \mathbb C). \end{align*} $$

Note that

$$ \begin{align*}\{\gamma \in [21435, 42531]_L \mid \mathrm{Des}_L(21435) \subseteq \mathrm{Des}_L(\gamma) \} = \{21435, 21543, 42531\}. \end{align*} $$

Since f is an $H_5(0)$ -module homomorphism, this equality implies that $c_\gamma = 0$ for all $\gamma \in [21435, 42531]_L \setminus \{21435,21543,42531\}$ . In addition, $c_{21543} = 0$ since

$$ \begin{align*}\pi_1 \pi_2 \cdot 21435 = 0 \quad \text{and} \quad \pi_1 \pi_2 \cdot f(21435) = c_{21543} \, 32541. \end{align*} $$

Hence, $f - c_{21435} \, \mathrm {id}$ is an $H_5(0)$ -module homomorphism such that

$$\begin{align*}(f - c_{21435} \mathrm{id}) (\gamma) = \begin{cases} c_{42531} 42531 & \text{if }\gamma = 21435,\\ 0 & \text{if }\gamma \in [21435, 42531]_L \setminus \{21435\}, \end{cases} \end{align*}$$

and therefore, $(f - c_{21435} \mathrm {id})^2 = 0$ . Since f is an idempotent, the possible values for $c_{21435}$ are $0$ or $1$ . Using the fact that f is an idempotent again, we have that $c_{42531} = 0$ . As a consequence, f is $0$ or $\mathrm {id}$ .

Based on the above discussion, we propose the following conjecture.

Conjecture 7.5. Let $P \in \mathsf {RSP}_n$ . Suppose that $\mathrm {sh}(\tau _P)$ is disconnected and does not contain any disconnected ribbon. Then, $\mathsf {M}_P$ is indecomposable.

7.3 Recovering $\mathsf {M}_P$ for $P \in \mathsf {RSP}_n$ from an $H_n(q)$ -module by specializing q to $0$

Let $q \in \mathbb C$ . The Hecke algebra $H_n(q)$ is the associative $\mathbb C$ -algebra with $1$ generated by $T_1, T_2, \ldots , T_{n-1}$ subject to the following relations:

$$ \begin{align*} T_i^2 &= (q-1) T_i + q \quad \text{for }1\le i \le n-1,\\ T_i T_{i+1} T_i &= T_{i+1} T_i T_{i+1} \quad \text{for }1\le i \le n-2,\\ T_i T_j &=T_j T_i \quad \text{if }|i-j| \ge 2. \end{align*} $$

Let $q \in \mathbb C$ be generic; that is, q is neither zero nor a root of unity. It is well known that $H_n(q)$ is isomorphic to the group algebra $\mathbb C[\mathfrak {S}_n]$ , and thus, the category of left finite dimensional $H_n(q)$ -modules is semisimple and there exists a ring isomorphism ([22, Section 3.2])

$$\begin{align*}\textbf{ch}_q: \bigoplus_{n\ge 0} \mathcal{G}_0(H_n(q)) \rightarrow \mathrm{Sym}, \quad [V^\lambda(q)] \mapsto s_\lambda. \end{align*}$$

Here, $\bigoplus _{n\ge 0} \mathcal {G}_0(H_n(q))$ is the Grothendieck ring of the tower of generic Hecke algebras equipped with addition and multiplication from direct sum and induction product, $\mathrm {Sym}$ is the ring of symmetric functions, and $V^\lambda (q)$ is the irreducible $H_n(q)$ -module attached to a partition $\lambda $ of size n. The explicit description of $V^{\lambda }(q)$ can be found in [21, p.7].

Let $P \in \mathsf {RSP}_n$ . Viewing q as an indeterminate, one may ask if $\mathsf {M}_P$ can be obtained from an $H_n(q)$ -module by specializing q to $0$ . However, it should be noted that the process of ‘specializing q to $0$ ’ depends on the choice of bases for the $H_n(q)$ -module under consideration, as illustrated in the example below.

Example 7.6. The irreducible $H_3(q)$ -module $V^{(2,1)}(q)$ has the underlying space $\mathbb C\{v_1, v_2\}$ , and the $H_3(q)$ -action defined by

$$ \begin{align*} \begin{cases} T_1 \cdot v_1 = -v_1, \\ T_2 \cdot v_1 = v_2, \end{cases} \quad \text{and} \quad\quad \begin{cases} T_1 \cdot v_2 = -q^2 v_1 + q v_2, \\ T_2 \cdot v_2 = qv_1 + (q - 1)v_2. \end{cases} \end{align*} $$

By the specialization $q = 0$ , we have the $H_3(0)$ -action on $\mathbb C\{v_1, v_2\}$ defined by

$$ \begin{align*} \begin{cases} \overline{\pi}_1 \cdot v_1 = -v_1, \\ \overline{\pi}_2 \cdot v_1 = v_2, \end{cases} \quad \text{and} \quad\quad \begin{cases} \overline{\pi}_1 \cdot v_2 = 0, \\ \overline{\pi}_2 \cdot v_2 = - v_2. \end{cases} \end{align*} $$

The resulting module is isomorphic to $\mathbf {T}^{+}_\unicode{x3b8} (\mathsf {M}_{P_1})$ , where $P_1 = \mathsf {poset}(\tau _0^{(2,1)}) \in \mathsf {RSP}_3$ .

However, if we choose the basis $\{w_1 := qv_1 - v_2, w_2 := (q^2 - q) v_1 - q v_2\}$ for $V^{(2,1)}(q)$ , then we have

$$ \begin{align*} \begin{cases} T_1 \cdot w_1 = w_2, \\ T_2 \cdot w_1 = -w_1, \end{cases} \quad \text{and} \quad\quad \begin{cases} T_1 \cdot w_2 = qw_1 + (q-1)w_2, \\ T_2 \cdot w_2 = -q^2 w_1 + q w_2. \end{cases} \end{align*} $$

By the specialization $q = 0$ , we have the $H_3(0)$ -action on $\mathbb C\{w_1, w_2\}$ defined by

$$ \begin{align*} \begin{cases} \overline{\pi}_1 \cdot w_1 = w_2, \\ \overline{\pi}_2 \cdot w_1 = -w_1, \end{cases} \quad \text{and} \quad\quad \begin{cases} \overline{\pi}_1 \cdot w_2 = -w_2, \\ \overline{\pi}_2 \cdot w_2 = 0. \end{cases} \end{align*} $$

The resulting module is isomorphic to $\mathbf {T}^{+}_\unicode{x3b8} (\mathsf {M}_{P_2})$ , where $P_2 = \mathsf {poset}(\tau _0^{(2,2)/(1)}) \in \mathsf {RSP}_3$ . It is worthwhile to remark that while $\mathbf {T}^{+}_\unicode{x3b8} (\mathsf {M}_{P_1})$ and $\mathbf {T}^{+}_\unicode{x3b8} (\mathsf {M}_{P_2})$ have the same quasisymmetric characteristic $\textbf {ch}_q([V^{(2,1)}(q)])$ , they are not isomorphic.

We expect that for $P \in \mathsf {RSP}_n$ , $\mathbf {T}^{+}_\unicode{x3b8} (\mathsf {M}_P)$ can be obtained from an $H_n(q)$ -module, whose image under $\textbf {ch}_q$ equals $K_P$ , by applying the specialization $q = 0$ to a suitable basis.