ON 2-TRANSITIVE SETS OF EQUIANGULAR LINES

Abstract

We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

1 Introduction

A set ${\mathcal L}$ of equiangular lines in a complex unitary vector space V is a set of $1$ -spaces that generates V such that the angle between any two members of ${\mathcal L}$ is constant. This is a notion that has arisen in various contexts, from combinatorics [14, 18] to quantum state tomography [16]. As in [11], this paper is concerned with sets of equiangular lines exhibiting a significant amount of symmetry.

Two sets of lines are equivalent if there is a unitary transformation sending one set to the other. The unitary automorphism group ${\mathbb A}{\mathrm {ut}}({\mathcal L})$ of ${\mathcal L}$ is the set of unitary transformations sending ${\mathcal L}$ to itself; the automorphism group ${\mathrm {Aut}}\,{\mathcal L}$ of ${\mathcal L}$ is the group of permutations of ${\mathcal L}$ induced by ${\mathbb A}{\mathrm {ut}}({\mathcal L})$ . The purpose of this note is to deal with a type of $2$ -transitive action of ${\mathrm {Aut}}\,{\mathcal L}$ not considered in [11].

Theorem 1.1. Let ${\mathcal L}$ be a $2$ -transitive set of equiangular lines in the complex unitary space V and such that the automorphism group of ${\mathcal L}$ has a regular normal subgroup. Let $|{{\mathcal L}}|=n$ , $\dim V=d$ and $1< d < n-1$ . Then one of the following occurs:

1. (i) $n=4$ and $d=2$ ;

2. (ii) $n=64$ and $d=8$ or $56$ ;

3. (iii) $n=2^{2m}$ and $d=2^{m-1}(2^m-1)$ or $2^{m-1}(2^m+1)$ for $m\geq 2$ ; or

4. (iv) $n=p^{2m}$ and $d=p^m(p^m-1)/2$ or $p^m(p^m+1)/2$ for a prime $p>2$ and $m\ge 1$ .

For each pair $(n,d)$ in (i)–(iv), there is a unique such set ${\mathcal L}$ up to equivalence.

We are assuming that ${\mathrm {Aut}}\,{{\mathcal L}}$ is finite and 2-transitive. Such a group has either a nonabelian quasi-simple socle (the so-called quasi-simple type) or it possesses a normal, regular subgroup (the so-called affine type). This note deals with the affine type. The quasi-simple type occurs in [11]. The case $n=d^2$ is completely settled in [22] producing (i), (ii) (and the case $n=3^2=d^2$ of (iv)), while the corresponding question over the reals is implicitly dealt with in [18] (producing (iii)). The assumption $1<d<n-1$ excludes degenerate examples (see [11]).

The proof of the theorem uses the classification of the finite $2$ -transitive groups (a consequence of the classification of the finite simple groups), together with mostly standard group theory and representation theory. We start with general observations concerning a 2-transitive line set ${\mathcal L}$ in a complex unitary space V. In Section 2.3, we show that ${\mathbb A}{\mathrm {ut}}({\mathcal L}) = Z({\mathrm {U}}(V)) G$ , where G is a finite group $2$ -transitive on ${\mathcal L}$ , and then that V is an irreducible G-module. The set-stabiliser $H=G_\ell $ of $\ell \in {{\mathcal L}}$ has a linear character $\lambda $ such that, if W is the module that affords the induced character $\lambda ^G$ , then $W=V\oplus V'$ for a second irreducible G-module $V'$ (Proposition 2.6(d)), which explains why $2$ -transitive line sets occur in pairs in the theorem. (See [11, page 3] for another explanation of this fact using Naimark complements.) Then we specialise to the case where ${\mathrm {Aut}}\,{\mathcal L}$ has a 2-transitive subgroup with a regular normal subgroup.

Section 2 contains group-theoretic background and Section 3 describes the examples in Theorem 1.1(iii) and (iv), while Section 4 contains the proof of the theorem. In the theorem, ${\mathbb A}{\mathrm {ut}}({\mathcal L})$ and ${\mathrm {Aut}}\,{{\mathcal L}}$ are as described in the following remark.

Remark 1.2. For ${\mathcal L}$ in Theorem 1.1, ${\mathbb A}{\mathrm {ut}}({\mathcal L}) =GZ$ , $Z=Z({\mathrm {U}}(V))$ where $G=E\rtimes S$ with a p-group E and $H =G_\ell $ , $\ell \in {\mathcal L}$ , is $Z(G)\times S$ , where $Z(G)=E\cap Z$ . In Section 4, we prove that the following statements hold for the various cases in the theorem:

1. (i) $E=Q_8,$ $|S|=3$ and $Z(G)=Z(E)$ has order $2;$

2. (ii) E is the central product of an extraspecial group of order $2^7$ with a cyclic group of order $4,$ $S\simeq {\mathrm {G}}_2(2)'\simeq {\mathrm {PSU}}(3,3)$ and $Z(G)=Z(E)$ has order $4;$

3. (iii) E is elementary abelian of order $2^{2m+1},$ $S\simeq {\mathrm {Sp}}(2m,2)$ and $Z(G)= E\cap Z$ has order $2;$ and

4. (iv) E is extraspecial of order $p^{2m+1}$ and exponent $p,\,$ $S\simeq {\mathrm {Sp}}(2m,p)$ and $Z(G)=Z(E)$ has order p.

2 Group theoretic background

Many facts of this section are basic and covered in the books of Aschbacher [1] and Huppert and Blackburn [10]. Our notation will follow the conventions of these references. We also need the classification of the $2$ -transitive finite groups. The groups of affine type are listed, for instance, in Liebeck [15, Appendix 1].

Lemma 2.1. Let G be a finite $2$ -transitive permutation group and $V\unlhd G$ an elementary abelian regular normal subgroup of order $p^t $ for a prime p. Identify G with a group of affine transformations $x\mapsto x^g+ c$ of $V=\mathbb F_p^t,$ where $g \in G_0$ and $0, c\in V$ . Then G is a semidirect product $V\rtimes G_0$ with $G_0\le {\mathrm {GL}}(V)$ , and one of the following occurs:

1. (i) $G_0\le \Gamma {\mathrm {L}}(1, p^t)$ ;

2. (ii) $G_0\unrhd {\mathrm {SL}}(s, q), q^s = p^t, s> 2$ ;

3. (iii) $G_0\unrhd {\mathrm {Sp}}(s, q), q^s = p^t$ ;

4. (iv) $G_0\unrhd {\mathrm {G}}_2(q)', q^6 = 2^t,$ where ${\mathrm {G}}_2(q)<{\mathrm {Sp}}(6,q)\le {\mathrm {Sp}}(t,2)$ ;

5. (v) $G_0$ is $ A_6\simeq {\mathrm {Sp}}(4, 2)'$ or $A_7, p^t=16$ ;

6. (vi) $G_0\unrhd {\mathrm {SL}}(2,3)$ with $t=2$ and $p^t= 5^2, 7^2, 11^2 $ or $23^2$ ;

7. (vii) $G_0\unrhd {\mathrm {SL}}(2,5)$ with $t=2$ and $p^t= 9^2, 11^2, 19^2, 29^2 $ or $59^2$ ;

8. (viii) $p^t = 3^4$ and $G_0 $ has a normal extraspecial subgroup Q of order $2^{1+4} $ such that $G_0= Q\rtimes S $ with $S\le {\mathrm {O}}^-(4,2)\simeq S_5$ and $|S|$ divisible by $5$ ;

9. (ix) $G_0'$ is ${\mathrm {SL}}(2, 13), p^t=3^6.$

2.1 Some indecomposable modules

Let U be an elementary abelian p-group (written additively) and $S\leq {\mathrm {Aut}}(U)$ , that is, we consider U as a faithful $\mathbb F_pS$ -module. We say that U is indecomposable if U is not the direct sum of two proper S-submodules. We are interested in modules with the following property.

Hypothesis (I). U has a trivial S-submodule $U_0 \neq 0$ , S acts transitively on the nontrivial elements of $V=U/U_0$ and the proper submodules of U lie in $U_0$ . The possible pairs $(S,V)$ are listed in Lemma 2.1 (S taking the role of $G_0$ ). The module U is an indecomposable module which extends a trivial module by V.

Lemma 2.2. Let U be an indecomposable $\mathbb F_pS$ -module satisfying (I) with $\dim U_0=1$ . Then $p=2$ and

1. (a) S has a normal subgroup $S_0$ and one of the following occurs:

   1. (1) $\dim V=2m$ , $m>1$ , $S_0\simeq {\mathrm {Sp}}(2a,2^b)'$ , $m=ab$ , or $S_0\simeq {\mathrm {G}}_2(2^b)'$ , $m=3b$ ; or

   2. (2) $\dim V=3$ , $S=S_0={\mathrm {SL}}(3,2)$ .

2. (b) The module U exists in case (a) and is unique as an $S_0$ -module.

3. (c) Let $S\simeq {\mathrm {Sp}}(2a,2^b)'$ , $m=ab$ , or $S\simeq {\mathrm {G}}_2(2^b)'$ , $m=3b$ . Then S has an embedding into a group $S^\star \simeq {\mathrm {Sp}}(2m,2)$ and U is the restriction of the unique $\mathbb F_2S^\star $ -module (satisfying (I)) to S.

Before we start the proof, we recall a few basic facts about group representations and cohomology. Let G be a finite group and V be an n-dimensional $FG$ -module associated with the matrix representation $D:G\to {\mathrm {GL}}(n,F)$ . Define the map $D^*: G\to {\mathrm {GL}}(n,F)$ by $D^*(g):= D(g^{-1})^t$ . With respect to $D^*$ , the space V becomes a G-module, the dual module $V^*$ of V.

We describe the connection of the existence of indecomposable modules with cohomology of degree $1$ and follow Aschbacher [1, Section 17]. Let G be a finite group and V a finite dimensional, faithful $\mathbb F_pG$ -module. A mapping $\delta : G\to V$ is called a derivation or $1$ -cocycle if $\delta (xy)=\delta (x)y+\delta (y)$ for all $x,y\in G$ . If $v\in V$ , then $\delta _v$ defined by $\delta _v(x)= v- vx$ is also a derivation. Such derivations are called inner derivations or $1$ -coboundaries. The set ${\mathrm {Z}}^1(G,V)$ of derivations and the set ${\mathrm {B}}^1(G,V)$ of inner derivations become elementary abelian p-groups with respect to pointwise addition. The factor group

$$ \begin{align*} {\mathrm{H}}^1(G,V)={\mathrm{Z}}^1(G,V)/{\mathrm{B}}^1(G,V) \end{align*} $$

is the first cohomology group of G with respect to V.

Suppose, V is a simple G-module. By Schur’s lemma, $K={\mathrm {End}}_{\mathbb F_pG}(V)$ is a finite field, say $\simeq \mathbb F_{p^e}$ , and $e\mid \dim V$ . For $\kappa \in K$ , $\delta $ a derivation, define $\delta \kappa :G\to V$ by $\delta \kappa (x)= \delta (x)\kappa $ . Then $\delta \kappa $ is a derivation and $\delta _v\kappa =\delta _{v\kappa }$ . So ${\mathrm {Z}}^1(G,V)$ , ${\mathrm {B}}^1(G,V)$ and ${\mathrm {H}}^1(G,V)$ become K-spaces.

We turn to Hypothesis (I) (S taking the role of G). By [1, (17.12)], we have the following assertions:

1. (i) there exists an $\mathbb F_pS$ -module with property (I) if and only if ${\mathrm {H}}^1(S,V^*)\neq 0$ ; and

2. (ii) every $\mathbb F_pS$ -module with property (I) is a quotient of a uniquely determined $\mathbb F_pS$ -module W with property (I) such that $\dim C_{W}(S) =\dim {\mathrm {H}}^1(S,V^*)$ .

If $V^*$ is simple then the module W in (ii) is even a $KS$ -module, where now $K={\mathrm {End}}_{\mathbb F_pS}(V^*)$ . So if U satisfies (I) and $\dim U_0=1$ , then there exists a hyperplane $W_0$ of $C_{W}(S)$ such that $U\simeq W/W_0$ . If $\dim _K{\mathrm {H}}^1(S,V^*)=1$ , then the multiplicative group of K acts transitively on the hyperplanes of $C_{W}(S)$ , that is, $U\simeq W/W_1$ for any hyperplane $W_1$ of $C_W(S)$ .

Proof of Lemma 2.2

Assume the existence of a module U as desired. Then S has no normal subgroup $N \neq 1$ with $(|N|,p)=1$ and $C_V(N)=0$ as otherwise, by [1, (24.6)], $U=[U,N]\oplus U_0$ is a G-decomposition. This excludes case (1) of Lemma 2.1 and forces $p=2$ (since $Z(S)$ contains an involution z with $C_V(z)=0$ if $p>2$ ).

So we have to consider cases (2)–(5) of Lemma 2.1 for S. Assume $\dim _{\mathbb F_2} V= 2^t$ . In cases (2)–(4), we have $S_0\unlhd S$ with $S_0 \simeq {\mathrm {SL}}(a,2^b)$ , $ab=t$ , $a>2$ , ${\mathrm {Sp}}(2a,2^b)'$ , $2ab=t$ , and ${\mathrm {G}}_2(2^b)'$ , $3b=t$ , and V is the defining $\mathbb F_{2^b}S_0$ -module. In case (2), we get assertion (a.2) by [12]. In cases (3) and (4), ${\mathrm {H}}^1(S_0,V^*)$ has dimension $1$ over $\mathbb F_{2^b}$ by [12]. It follows that a module with property (I) and $\dim U_0=1$ exists and is unique up to isomorphism. We get assertions (a) and (b) once we exclude case (5). So assume $S\simeq {\mathrm {A}}_7$ , U is a $5$ -dimensional $\mathbb F_2S$ -module, $U/U_0$ is simple and $\dim U_0 =1$ for $U_0=C_U(S)$ . There are $16$ hyperplanes in U that intersect $U_0$ trivially. A permutation representation of S of degree $\leq 16$ has degree $1,7$ or $15$ . Hence, $U_0$ has an S-invariant complement in U and U is decomposable. This excludes case (5).

For (c), note that $S\simeq {\mathrm {Sp}}(2a,2^b)'$ , $ab=m$ , is a subgroup of $S^\star = {\mathrm {Sp}}(2m,2)\simeq {\mathrm {O}}(2m+1,2)$ [9, Hilfssatz 1] and so is $S\simeq {\mathrm {G}}_2(2^b)'$ , $3b=m$ [15, page 513]. The indecomposable $S^\star $ -module U is the ${\mathrm {O}}(2m+1,2)$ -module [17, pages 55, 143]. As S acts transitively on $V\simeq U/U_0$ , we see that U is indecomposable as an S-module.

2.2 On representations of extraspecial groups

A finite, nonabelian p-group E (p a prime) is extraspecial if $Z(E)=E'=\Phi (E)$ has order p (these groups have many other names, such as ‘Heisenberg groups’, ‘Weyl–Heisenberg groups’ and ‘generalised Pauli groups’). We consider the following property.

Hypothesis (E). Let p be a prime and $m\geq 1$ an integer. If $p>2$ , then E is an extraspecial group of order $p^{1+2m}$ and exponent p and if $p=2$ , then E is the central product of an extraspecial group of order $2^{1+2m}$ with a cyclic group of order $4$ .

Assume Hypothesis (E) and let $A=\{ \alpha \in {\mathrm {Aut}} (E) \mid \alpha _{Z(E)}=1_{Z(E)} \}$ be the centraliser of $Z(E)$ in the automorphism group. Then (see [7, 21]),

(2.1) $$ \begin{align} A/{\mathrm{Inn}}(E) \simeq {\mathrm{Sp}}(2m,p). \end{align} $$

Denote by $\zeta _k=\exp ( 2\pi i/k)$ a primitive kth root of unity. Assertions (a) and (b) of the next Lemma are [1, (34.9)] and [10, Satz V.16.14], whereas the last assertion follows from [21, Theorem 1].

Lemma 2.3. Assume Hypothesis (E) and let U be a $p^m$ -dimensional complex space. Set $Z(E)=\langle z\rangle $ .

1. (a) In the case $p=2$ , there exist precisely two faithful, irreducible representations $D_j: E\to {\mathrm {GL}}(U)$ , $j=1,3$ , and $D_j(z)=\zeta _4^j \cdot 1_U$ . Every faithful, irreducible representation of E is of this form.

2. (b) In the case $p>2$ , there exist precisely $p-1$ faithful, irreducible representations $D_j: E\to {\mathrm {GL}}(U)$ , $1\leq j\leq p-1$ , and $D_j(z)=\zeta _p^j \cdot 1_U$ . Every faithful, irreducible representation of E is of this form.

For each j, there is an automorphism $\gamma _j$ of E such that $D_j$ can be defined by $D_j(e)= D_1(e{\gamma _j})$ for all $e\in E,$ so $D_j(E)=D_1(E)$ .

2.3 Basic properties of $2$ -transitive line sets

In this subsection, ${\mathcal L}$ denotes a $2$ -transitive set of n equiangular lines in a complex unitary space V of dimension $d<n$ . Let K be the kernel of the permutation action of ${\mathbb A}{\mathrm {ut}}({\mathcal L})$ on ${\mathcal L},$ which clearly contains $Z :=Z({\mathrm {U}}(V))$ .

Lemma 2.4. We have $K= Z$ .

Proof. Let $g\in K$ . Let m be the minimal number of nonzero $a_i$ in a dependency relation $\sum _i a_iv_i=0$ , $\langle v_i\rangle \in {\mathcal L}$ . Apply g to obtain another dependency relation $\sum _i k_ia_iv_i=0$ with the same m nonzero $k_ia_i$ ; these relations must be multiples of one another by minimality. Thus, restricting to nonzero $a_i$ produces constant $k_i$ .

Any two different members $\langle v_i\rangle , \langle v_j\rangle $ of ${\mathcal L}$ occur with nonzero coefficients in such a relation. Then g acts on all members of ${\mathcal L}$ with the same scalar, and so is a scalar transformation since ${\mathcal L}$ spans V.

Lemma 2.5. There is a finite group G such that ${\mathbb A}{\mathrm {ut}}({\mathcal L}) =GZ$ .

Proof. By [1, (33.9)], $D={\mathbb A}{\mathrm {ut}}({\mathcal L})'$ is finite. Let $G\leq {\mathbb A}{\mathrm {ut}}({\mathcal L})$ be a finite group such that $D\leq G$ and $GZ/Z$ has maximal order in ${\mathrm {Aut}}\,{\mathcal L}= {\mathbb A}{\mathrm {ut}}({\mathcal L}) /Z$ . Suppose $GZ<{\mathbb A}{\mathrm {ut}}({\mathcal L})$ . Pick $h\in {\mathbb A}{\mathrm {ut}}({\mathcal L}) -GZ$ . Then $h^m\in Z$ for some integer m, so there is $z\in Z$ such that $h^m=z^{-m}$ . Since $[G,hz]\subseteq D\le G$ , we get $|\langle G, hz\rangle |<\infty $ and $GZ/Z< \langle G, h\rangle Z/Z= \langle G, hz\rangle /Z$ , a contradiction.

Proposition 2.6. Let G be as in Lemma 2.5 and let $H=G_\ell $ , $\ell \in {{\mathcal L}}$ , be the stabiliser of a line. Let $\lambda $ be the linear character of H afforded by $\ell $ . Then:

1. (a) V is simple and a constituent of the module W which affords $\lambda ^G$ ;

2. (b) $W=V\oplus V'$ with a simple module $V'$ inequivalent to V;

3. (c) V and $V'$ as H-modules afford $\lambda $ with multiplicity $1$ ; and

4. (d) there is a set ${\mathcal L}'$ of n lines of $V'$ on which G acts $2$ -transitively if $d< n-1$ .

Proof. By 2-transitivity, $G=H\cup HtH$ for $t\in G-H$ . Assume that $V=V_1\oplus \cdots \oplus V_r$ for simple G-modules $V_i$ . Let $\chi _i$ be the character of $V_i$ .

Let $\ell =\langle v\rangle $ . If $v=v_1+\cdots +v_r$ with $v_i\in V_i$ , then each $v_i\ne 0$ since $\langle {\mathcal L} \rangle =V$ . As $\lambda (h)v=\lambda (h)v_1+\cdots +\lambda (h)v_r$ for $h\in H$ , $\lambda $ is a constituent of $(\chi _i)_H$ . By Frobenius Reciprocity, each $\chi _i$ is a constituent of $\lambda ^G$ .

We claim that $\lambda ^G=\psi _1+\psi _2$ for distinct irreducible characters $\psi _i$ of G. For, by Mackey’s theorem [10, Satz V.16.9], $(\lambda ^G)_H=((\lambda ^{1^{-1}}) _{H\cap H^1})^H + ((\lambda ^{t^{-1}} )_{H\cap H^t}) ^H. $ By Frobenius Reciprocity, $(\lambda ^G, \lambda ^G)= (\lambda , (\lambda ^G)_H) = 1+(\lambda ,((\lambda ^{t^{-1}} )_{H\cap H^t}) ^H)$ and $(\lambda ,((\lambda ^{t^{-1}} )_{H\cap H^t}) ^H)= (\lambda _{H\cap H^t},(\lambda ^{t^{-1}} )_{H\cap H^t})$ . Hence, $(\lambda ^G, \lambda ^G)=1$ or 2. If $\lambda ^G$ is irreducible, then each $\chi _i=\lambda ^G$ , so $d=r\lambda ^G(1)=r|{\mathcal L}|\ge n$ . This contradiction proves the claim. By Frobenius Reciprocity, $(\lambda ,(\psi _i)_H)=1$ for $i=1,2$ . Then (a)–(c) follow if $r=1$ .

We now assume $r>1$ . Each $\chi _i$ is in $\{ \psi _1,\psi _2 \}.$ If $ \{ \chi _1,\chi _2 \}=\{ \psi _1,\psi _2 \}$ , then we would have $d\ge \chi _1(1)+\chi _2(1)= \lambda ^G(1)=|{\mathcal L}|$ , which is not the case.

Since $\psi _1\ne \psi _2$ , we are left with the possibility $\chi _1=\chi _2\in \{ \psi _1,\psi _2 \}$ , say $\chi _i=\psi _1$ . Let $\phi \colon V_1\to V_2$ be a G-isomorphism. Since $\lambda $ has multiplicity $1$ in $\psi _1$ , the morphism $\phi $ sends the unique submodule of $(V_1)_H$ affording $\lambda $ to the unique submodule of $(V_2)_H$ affording $\lambda $ . Thus, $v_1\phi =av_2$ with $a\in \mathbb C^*$ . Then

$$ \begin{align*}\langle v_1g+v_2g\mid g\in G \rangle =\langle v_1g+a^{-1}v_1\phi g\mid g\in G \rangle =V_1(1+ a^{-1}\phi ),\end{align*} $$

showing $\langle {{\mathcal L}} \rangle \subseteq V_1(1+ a^{-1}\phi )\oplus V_3\oplus \cdots \oplus V_r$ . This contradicts the fact that ${\mathcal L}$ spans V.

For (d), note that by (c), $V'$ contains an H-invariant $1$ -space $\ell '$ . Then $\ell ' G$ is a 2-transitive line set of size n since $\dim V' =n-d>1$ and since H is maximal in G.

Remark 2.7. $\lambda $ is a nontrivial character for $1< d< n-1$ (since $((1_H)^G,1_G)=1$ by Frobenius Reciprocity).

3 Examples of $2$ -transitive line sets

In this section, we describe the examples listed in Theorem 1.1. See [8, 22] for Theorem 1.1(i) and (ii).

Example 3.1 (for Theorem 1.1(iii))

Let $m>1$ and let $E = \mathbb F_2^{2m+1}$ . Then E is an ${\mathrm {O}}(2m+1,2)$ -space with radical R [17, pages 55, 143]. Then $S:= {{\mathrm {O}}(2m+1,2)}\simeq {\mathrm {Sp}}(2m,2)={\mathrm {Sp}}(E/R)$ is transitive on the $d:= 2^{m-1}{(2^m-1)}$ hyperplanes of E of type ${\mathrm {O}}^-(2m,2)$ and on the $2^{m-1}(2^m+1)$ hyperplanes of type ${\mathrm {O}}^+(2m,2)$ [17, page 139]. Label the standard basis elements of $V= \mathbb C ^d$ as $v_M$ with M ranging over the first of these sets of hyperplanes. Let S act on this basis as it does on these hyperplanes. This action is 2-transitive (as observed implicitly for line sets in [18] and first observed in [5]), so the only irreducible S-submodules of V are $\langle \bar v\rangle $ and $\bar v^\perp $ , where $\bar v:=\sum _Mv_M$ .

Each such M is the kernel of a unique character $\lambda _M\colon E\to \{\pm 1\}$ . Let $e\in E$ act on V by $v_Me:= \lambda _M(e)v_M$ for each basis vector $v_M$ . If $1\ne r\in R $ , then $\lambda _M(r)=-1$ since $r\notin M$ , so r acts as $-1$ on V. If $e\in E$ and $h\in S$ , then $ (\bar v e )h= \bar v h\cdot h^{-1}eh = \bar v e^h$ , so S acts on $\langle \bar v\rangle E$ , a set of 1-spaces of V. Since S is irreducible on $\bar v^\perp $ , the set $\langle \bar v\rangle E=\langle \bar v\rangle {ES}$ spans V and $\langle \bar v\rangle $ is the only 1-space fixed by S. In particular, $\langle \bar v\rangle $ affords the unique involutory linear character $\lambda $ of $H=R\times S$ whose kernel is S. Clearly, $(E/R)\rtimes S$ acts 2-transitively on the $n=2^{2m}$ cosets of S. These are the d-dimensional examples in Theorem 1.1(iii). The $2^{m-1}{(2^m+1)}$ hyperplanes of type ${\mathrm {O}}^+(2m,2)$ produce similarly the $(n-d)$ -dimensional examples.

Example 3.2 (For Theorem 1.1(iv))

Let $p>2$ be a prime, m a positive integer and E an extraspecial group of order $p^{1+2m}$ and exponent p. Using Lemma 2.3, we consider E as a subgroup of ${\mathrm {U}}(W)$ , W a complex unitary space of dimension $p^{m}$ . By [2], the normaliser of E in ${\mathrm {U}}(W)$ contains a subgroup $G=E\rtimes S$ , $G/E\simeq {\mathrm {Sp}}(2m,p)$ inducing ${\mathrm {Sp}}(2m,p)$ on $E/Z(E)$ , with $ES$ acting 2-transitively on the $n=p^{2m}$ cosets of $H=Z(E)\times S$ . Moreover, $Z(S)=\langle z\rangle $ has order 2, and $W=W_+\perp W_-$ for the eigenspaces $W_+$ and $ W_-$ of z (with $\dim W_-= (p^m-\varepsilon )/2$ for $\varepsilon \in \{ \pm 1\}$ , $p^m \equiv \varepsilon \pmod {4}$ ); these are irreducible S-modules (Weil modules) [2, 6].

Let U be one of these eigenspaces, say of dimension d. As $G/E\simeq S$ , we can consider U as a G-module. Define $V:= W\otimes U^* \subset W\otimes W^*$ ( $U^*$ dual to U). If $\chi $ is the character of S on U, then $\chi \bar \chi $ is the character of S on $U\otimes U^*$ . Trivially, $(\,\chi \, \bar \chi ,1_S)=(\,\chi ,\chi )=1$ , so there is a unique 1-space $\langle v_0\rangle $ in $U\otimes U^*$ (and hence in V) fixed pointwise by S (and it is the only 1-space fixed by the group S). In particular, $\langle v_0\rangle $ affords a nontrivial linear character $\lambda $ of H with kernel S. Since E is irreducible on W while S is irreducible on $U^*$ , the set $\langle v_0\rangle {ES}$ spans V. These are the examples in Theorem 1.1(iv).

Lemma 3.3. Let p be a prime, $m\geq 1$ an integer and $G=ES$ as in Example 3.1 if $p=2$ and as in Example 3.2 if $p>2$ . Let ${\mathcal L}$ be a line set of size $n=p^{2m}$ in a complex unitary space V with $1<\dim V< n-1$ such that $G\leq {\mathbb A}{\mathrm {ut}}({\mathcal L})$ induces a $2$ -transitive action on ${\mathcal L}$ . Then ${\mathcal L}$ is equivalent to a line set of Example 3.1 or 3.2.

Moreover, if $\lambda $ is a linear character of $Z(G)\times S$ , $\ker \lambda =S$ , then every constituent of the module associated with $\lambda ^G$ contains a G-invariant line set satisfying the assumptions of this lemma.

Proof. For $i=1,2$ , let ${\mathcal L}_i\subseteq V_i$ be line sets in complex unitary spaces and let $G_i=E_i\rtimes S_i \leq {\mathrm {U}}(V_i)$ , $S_i\simeq {\mathrm {Sp}}(2m,p)$ be isomorphic groups as in the examples with a $2$ -transitive action on ${\mathcal L}_i$ . Let $\ell _i\in {\mathcal L}_i$ and $H_i=(G_i)_{\ell _i}$ . We assume that one of the line sets belongs to an example and, arguing by symmetry, we can also assume $1< \dim V_i\leq n/2$ , $i=1,2$ .

Claim. ${\mathcal L}_1$ is equivalent to ${\mathcal L}_2$ . By Proposition 2.6 and Remark 2.7, the representation $\lambda _i$ of $H_i$ on $\ell _i$ is a nontrivial linear character of $H_i$ . We have $H_i=Z_i\times S_i$ , $Z_i=Z(G_i)$ . Let $\alpha : G_1\to G_2$ be an isomorphism.

Case $p>2$ . The group $S_i$ is a representative of the unique class of complements of $E_i$ in $G_i$ (note that $S=C_{G}(Z(S))$ and $Z(S)$ is a Sylow 2-subgroup of $E\rtimes Z(S)\unlhd G$ ). So we can assume $H_2=H_1\alpha $ , $S_2=S_1\alpha $ . We also can assume $S_i=\ker \lambda _i$ by Lemma 4.1 below. By Lemma 2.3, there exists an automorphism $\gamma $ of $G_1$ such that $\lambda _1(z)=\lambda _2(z\gamma \circ \alpha )$ for $z\in Z$ . So replacing, if necessary, $\alpha $ by $\gamma \circ \alpha $ , we may assume that $\lambda _1(z)=\lambda _2(z\alpha )$ holds. Define a representation $D: G_1\to {\mathrm {GL}}(V_2)$ by

$$ \begin{align*} v D(g)=v (g\alpha), \quad v\in V_2,\ g\in G_1. \end{align*} $$

Let W be the module associated with the induced character $\lambda _1^{G_1}$ . By Proposition 2.6, both $G_1$ -modules are isomorphic to the same irreducible submodule of W, that is, $V_1\simeq V_2$ . Hence, there exists a $G_1$ -morphism $\phi : V_1\to V_2$ with $\ell _1\pi = \ell _2$ ( $\lambda _1$ has multiplicity $1$ in $V_1$ and $V_2$ ). The claim holds for $p>2$ .

Case $p=2$ . Assume first $m>2$ . Then $S_2$ and $S_1\alpha $ are complements of $E_2$ in $G_2$ . By [1, (17.7)], there exists $\beta \in {\mathrm {Aut}}(G_2)$ with $S_2=(S_1\alpha )\beta $ . So replacing $\alpha $ , if necessary, by $\alpha \circ \beta $ , we can assume $H_1\alpha =H_2$ and $S_1\alpha =S_2$ . Note that H has precisely one nontrivial linear character. Now arguing as in the case $p>2$ , we see that ${\mathcal L}_1$ and ${\mathcal L}_2$ are equivalent. In the case $m=2$ , replace $S_i$ by $S_i'$ . Then the argument from case $m>2$ carries over and shows the equivalence of ${\mathcal L}_1$ and ${\mathcal L}_2$ . The first assertion of the lemma holds and the second follows from the preceding discussion. □

4 Proof of Theorem 1.1 and automorphism groups

In this section, p is a prime and ${{\mathcal L}}$ denotes a set of $n=p^t$ equiangular lines in a complex unitary space V of dimension d with $1< d< n-1$ . By the assumptions of Theorem 1.1 and the results of Section 2.3, there exists a finite group $G\leq {\mathbb A}{\mathrm {ut}}({\mathcal L})$ with a $2$ -transitive action on ${\mathcal L}$ . Set $Z=Z(G)$ . Then $G/Z$ has a regular normal subgroup and V is a simple G-module. We assume $n\neq 4$ . As for $n=4$ , the results in [22] imply assertion (i) of Theorem 1.1. It suffices to assume that no proper subgroup of $G/Z$ has a $2$ -transitive action on ${{\mathcal L}}$ and that no subgroup of ${\mathbb A}{\mathrm {ut}}({\mathcal L})$ , which covers the quotient $GZ/Z$ , has order $<|G|$ . We set $H=G_{\ell }$ , $\ell \in {{\mathcal L}}$ . Then the character/representation $\lambda : H\to {\mathrm {U}}(\ell )$ of H on $\ell $ is nontrivial by Remark 2.7. Observe that there is some flexibility in the choice of G: generators of G can be adjusted by scalars. We show that G can be chosen such that $G\leq \tilde G$ where $\tilde G$ is a group which is used to construct a line set in Examples 3.1 and 3.2.

Lemma 4.1. We may assume $G=E\rtimes S$ , $H=Z\times S$ , where S is the kernel of the action of H on $\ell $ . Moreover, $Z\leq E$ and one of the following occurs:

1. (a) $p=2$ , E is an elementary abelian $2$ -group, $|Z|=2$ and E as an S-module satisfies Hypothesis (I); or

2. (b) $t=2m$ , E satisfies Hypothesis (E) and $E/Z(E)$ is a simple S-module.

Proof. Let M be the pre-image of the regular, normal subgroup of $G/Z$ . Since $M/Z$ is abelian, we have $M=E\times Z_{p'}$ with a Sylow p-subgroup E of M and $Z_{p'}$ is the largest subgroup of Z with an order coprime to p. Let L be the kernel of $\lambda $ .

We may assume that $E=M$ , $Z\leq E$ and $S=L$ is a complement of Z in H. Clearly, $Z\leq H\cap M$ and $L\cap Z=1$ . As $H/L$ is cyclic, we can choose $c\in H$ such that $H=\langle c,L\rangle $ . Pick $\omega \in \mathbb C$ of norm $1$ such that $S=\langle \omega c,L\rangle $ has a trivial action on $\ell $ . Then ${\tilde {G}=ES}$ is $2$ -transitive on ${\mathcal L}$ . Moreover, $S\cap E\leq S\cap (\tilde {G}_\ell \cap E) \leq S\cap Z({\mathrm {U}}(V)) =1$ . Since ${Z\geq Z\cap E= Z(\tilde {G})\cap E=Z(\tilde {G})}$ and $G/Z\simeq \tilde {G}/Z(\tilde {G})$ , we get $|\tilde {G}|\leq |G|$ . So we may assume $G=\tilde {G}$ and $H=(E\cap Z)\times S$ . In particular, $Z\leq E$ .

Assume first that E is abelian. Set $\Omega =\langle e\in E \mid |e|=p\rangle $ . This group is a characteristic elementary abelian subgroup of E. If $\Omega \leq Z$ , then E is cyclic, and $S\neq 1$ is a $p'$ -group (isomorphic to a subgroup of ${\mathrm {Aut}}(E)$ of order $p-1$ ). By Remark 2.7, $Z\neq 1$ . This contradicts [1, (23.3)] (on automorphism groups of cyclic groups).

So $E = \Omega Z$ and, by the minimal choice of G, we obtain $E=\Omega $ . If Z has an S-invariant complement $E_0$ in E, then, by induction, $G=E_0S$ contradicting $Z\neq 1$ . So $1< Z <E$ is the unique composition series of E as an S-module and assertion (a) follows as Z is cyclic.

Assume now that E is nonabelian. If N were a characteristic, normal, abelian subgroup of E of rank $\geq 2$ , then $1< NZ/Z \leq E/Z$ would be an S-invariant series. By our minimal choice $N=E$ , this is absurd. So E is of symplectic type and therefore, by [1, (23.9)], $E=C\circ E_1$ where E is extraspecial or $=1$ and C is cyclic or $p=2$ or C is a generalised quaternion group, a dihedral group or a semidihedral group of order $\geq 16$ .

Suppose $p>2$ . By [1, (23.11)], E is extraspecial of exponent p. So assertion (b) follows for $p>2$ .

Suppose finally $p=2$ . A standard reduction (see for instance [19, Lemma 5.12]) shows that E contains a characteristic subgroup F such that F is extraspecial of order $2^{1+2m}$ or satisfies hypothesis (E). By our choice of G, we have $E=F$ as $t=2m>2$ . If E is extraspecial, then S cannot act transitively on the nontrivial elements of $E/Z(E)$ as there are cosets modulo $Z(E)$ of elements of order $4$ as well as cosets of elements of order $2$ . So assertion (b) holds for $p=2$ .

By Lemma 4.1, we distinguish the cases E abelian ( $p=2$ ), E nonabelian, $p>2$ , and E nonabelian, $p=2$ . Then Lemmas 4.2 and 4.3 complete the proof of Theorem 1.1. The proof of Lemma 4.2 is very similar to the proof of Lemma 3.3.

Lemma 4.2. The following assertions hold.

1. (a) If E be abelian, then Theorem 1.1(iii) holds.

2. (b) If E be nonabelian and $p>2$ , then Theorem 1.1(iv) holds.

Proof. If E is abelian, Lemma 2.2 applies. Case (a.2) of this lemma does not occur. Let $G=E\rtimes S$ , $S\simeq {SL}(3,2)$ , $Z=C_E(S)$ and $E/Z$ be the natural S-module. A simple E-module in V affords a nontrivial character $\chi $ of E and its kernel $E_ \chi $ is a hyperplane intersecting Z trivially. There are precisely $8$ such hyperplanes. The group S acts transitively on these hyperplanes (otherwise, as the smallest degree of a nontrivial permutation representation of S is $7$ , S would fix one of these hyperplanes and E would not be an indecomposable S-module). Hence, $\dim V\geq 8 =n$ , a contradiction.

So there exists an embedding $\iota : G\to \tilde G$ , $\tilde G= \tilde E\rtimes \tilde S$ , $\tilde S\simeq {\mathrm {Sp}}(2m,p)$ with $\tilde E= E\iota $ , $S\iota \leq \tilde S$ . This follows from (c) of Lemma 2.2 if $p=2$ and for $p>2$ , it is clear by (2.1). The linear character $\tilde \lambda $ of $H\iota $ defined by

(4.1) $$ \begin{align} \tilde \lambda (h\iota )=\lambda( h ),\quad h\in H, \end{align} $$

has a unique extension to $\tilde H= Z\iota \times \tilde S$ such that $\ker \tilde \lambda = \tilde S$ . Let $\tilde W$ be the module associated with the induced character $(\tilde \lambda )^{\tilde G}$ . By Proposition 2.6 and Lemma 3.3, we have a decomposition into simple $\tilde G$ -modules $\tilde W= \tilde V\oplus \tilde V'$ and both modules contain $\tilde G$ -invariant line sets. We turn $\tilde W$ into a G-module by

$$ \begin{align*} \tilde w\cdot g= \tilde w (g\iota),\quad \tilde w \in \tilde W, \ g\in G. \end{align*} $$

By Mackey’s theorem [10, Satz V.16.9] and (4.1),

$$ \begin{align*} ((\tilde \lambda)^{\tilde G})_G = ((\tilde \lambda)_{{\tilde H}\cap G\iota})^G =(\lambda_{H})^G. \end{align*} $$

So $\tilde W$ as a G-module affords $\lambda ^G$ . Then by Proposition 2.6, V is isomorphic to $\tilde V$ or $\tilde V'$ . Say $V\simeq \tilde V$ . An isomorphism $\phi : V\to \tilde V$ maps the line set ${\mathcal L}$ onto ${\mathcal L}\phi $ such that $\ell $ and $\ell \phi $ both afford as H-spaces the character $\lambda $ . However, $\tilde V$ contains a $\tilde G$ -invariant line set containing a line affording $\tilde \lambda $ . Thus, by (4.1) and Proposition 2.6, ${\mathcal L}\phi $ is this $\tilde G$ -invariant line set. Using Lemma 3.3 again completes the proof.

Lemma 4.3. Let E be nonabelian and $p=2$ . Then (i) or (ii) of Theorem 1.1 hold.

Proof. By Proposition 2.6, we may assume $d=\dim V \leq n/2= 2^{2m-1}$ . As E satisfies Hypothesis (E), S is isomorphic to a subgroup of ${\mathrm {Sp}}(2m,2)$ (see (2.1)). By Lemma 2.1 and by the minimal choice of G, we have $H/Z(H)\simeq {\mathrm {SL}}(2,2^m)$ or $\simeq {\mathrm {G}}_2(2^b)'$ and $b=m/3$ . Let $V=V_1\oplus \cdots \oplus V_\ell $ , a decomposition into irreducible E-modules. Clearly, all $V_i$ are faithful E-modules, in particular, $d=2^m\ell $ . A generator of Z induces the same scalar on each $V_i$ as the eigenspaces of this generator are G-invariant. Lemma 2.3 shows that all $V_i$ ’s are pairwise isomorphic. If $\ell =1$ , then $n=2^{2m} =d^2$ and an application of the main result of [22] proves the assertion of the lemma.

So assume $\ell>1$ . Denote by D the representation of G afforded by V and apply [10, Satz V.17.5]. Then $D(g)=P_1(g)\otimes P_2(g)$ where the $P_i$ terms are irreducible projective representations of G and $P_2$ is also a projective representation of $S\simeq G/E$ of degree $\ell $ . Denote by $m_S$ the minimal degree of a nontrivial projective representation of S. By [10, Satz V.24.3], $m_S$ is the minimal degree of a nontrivial, irreducible representation of the universal covering group of S. We have $m_S=2^m-1$ for $S\simeq {\mathrm {SL}}(2,2^m)$ , $m>3$ [20, Table 3], [13], $m_S=2^m-2^b$ for $S\simeq {\mathrm {G}}_2(2^b)'$ , $m=3b$ , $b\neq 2$ [20, Table 3], [13], $m_S=2$ for $S\simeq {\mathrm {SL}}(2,4)$ , $m=2$ [4], and $m_S=12$ for $S\simeq {\mathrm {G}}_2(4)$ , $m=12$ [4]. Since $m_S2^m\leq d \leq 2^{2m-1}$ , only the last two cases may occur.

For $S\simeq {\mathrm {G}}_2(4)$ , degree $12$ is the only degree of a nontrivial, irreducible, projective representation of degree $\leq 64$ . By Proposition 2.6, there exists an irreducible G-module $V'$ such that $\dim V'= 2^{12}-d =64\cdot 52$ and $52$ is the degree of of an irreducible, projective representation of S, a contradiction.

Assume finally $m=2$ . It follows from [7, Theorem 4] that there exists a group $G=E\rtimes S$ , $S\simeq {\mathrm {SL}}(2,4)$ , and this group is unique up to isomorphism. Using GAP or Magma, one can compute characters of G. For $H=Z(E)\times S$ , there exist precisely two linear characters of H with kernel S. For any such character $\lambda $ , the induced character $\lambda ^G$ is irreducible, which rules out this possibility too.

4.1 Automorphism groups

Proof of Remark 1.2

For cases (i) and (ii), we refer to [8, 22]. For the remaining two cases, we have, by Theorem 1.1, a finite subgroup $G=E\rtimes S\leq {\mathbb A}{\mathrm {ut}}({\mathcal L})$ , with $|E/(E\cap Z)|=p^{2m}$ , $Z=Z({\mathrm {U}}(V))$ and $S\simeq {\mathrm {Sp}}(2m,p)$ . The assertions follow in cases (iii) and (iv) if $E/(E\cap Z)$ is normal in ${\mathrm {Aut}}\,{\mathcal L}$ , that is, if ${\mathrm {Aut}}\,{\mathcal L}$ has a regular, abelian normal subgroup. Suppose ${\mathrm {Aut}}\,{\mathcal L}$ has a nonabelian simple socle. Then, by the classification of the $2$ -transitive groups (see [3]), ${\mathrm {Aut}}\,{\mathcal L}$ is at least triply transitive. In that case, the application of Proposition 2.6 (to a point stabiliser) forces $\dim V= d=n-1$ , a contradiction.