We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.

Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{{\mathcal{O}}}$ of any conjugacy class ${\mathcal{O}}$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_{{\mathcal{O}}}}$, where ${\mathcal{O}}$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H,H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

This note proves Cellini’s conjecture that, in a Coxeter system (W,S) with reflections T, the T-increasing paths in W are self-avoiding. Here, a T-increasing path is a sequence v,t1v,…,tn⋯t1v in W with ti∈T and t1≺⋯≺tn in a reflection order ⪯ of T.

functional identities for the classical rogers dilogarithm associated to simply laced dynkin diagrams are proved uniformly in the setting of $y$-systems related to clusters.

We describe the Zariski closure of the geometric realization of Coxeter groups. When the Coxeter system is irreducible and the Tits form B non-positive and non-degenerate, this Zariski closure is equal to the orthogonal group of B.

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ and $h$ is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of $\Phi$ are included.