Truncations and extensions of the Brauer-Chen algebra

TL;DR

This paper explores the representation theory of a generalized algebra related to complex reflection groups, introducing extensions and deformations that broaden understanding of algebraic structures in this domain.

Contribution

It determines the generic representation theory of a key quotient of the Brauer-Chen algebra and introduces new algebra extensions with monodromic deformations.

Findings

Representation theory of the algebra's quotient is characterized.

Extensions of the algebra are defined and shown to admit deformations.

The work generalizes classical algebraic structures to complex reflection groups.

Abstract

The Brauer-Chen algebra is a generalization of the algebra of Brauer diagrams to arbitrary complex reflection groups, that admits a natural monodromic deformation. We determine the generic representation theory of the first non trivial quotient of this algebra. We also define natural extensions of this algebra and prove that they similarly admit natural monodromic deformations.

Equations

$$|W|+\sum _{{\mathcal{A}}_0\in \mathcal{A}/W}|{\mathcal{A}}_0|\times |W|/|W_{H_0}|$$

$$e_{H_1}{e}_{H_2}=\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s\right)e_{H_2}=e_{H_1}\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s\right)$$

$$(1{)}^{\prime \prime }s{e}_H=e_{H}s=e_H\text{mbox}ifs\in \mathcal{R},s(H)=H\text{mbox}andH\text{mbox}and\mathrm{Ker}(s-1)\text{mbox}are(distinctand)nottransverse.$$

$$\omega =\sum _{H\in \mathcal{A}}\left(\left(\sum _{Ker(s-1)=H}{\mu }_{s}s\right)-e_H\right){\omega }_H\in {\Omega }^1(X)\otimes B{r}^0(W)$$

$$[t_{H_0},t_Z]=[{\phi }_{H_0}-e_{H_0},{\phi }_Z-\sum _H{e}_H]=-[{\phi }_{H_0},\sum _H{e}_H]-[e_{H_0},{\phi }_Z-\sum _H{e}_H].$$

$$e_{H_0}\sum _H{e}_H=\sum _H{e}_{H_0}{e}_H=\sum _H{e}_{H_0}\sum _{s(H)=H_0}{\mu }_{s}s=e_{H_0}\sum _{s\in \mathcal{R};\mathrm{Ker}(s-1)\supset Z}{\mu }_{s}s=e_{H_0}{\phi }_Z$$

$$[e_{H_0},{\phi }_Z-\sum _H{e}_H]=0$$

$$(H_1,\dots ,H_r)=(\underset{u_1}{\underbrace{J_1,\dots ,J_1}},\dots ,\underset{u_k}{\underbrace{J_k,\dots ,J_k}})$$

$$w{t}_H=t_{w(H)}w,\left[t_H,\sum _{H^{\prime }\supset Z}{t}_{H^{\prime }}\right]=0$$

$$\left\{\begin{array}[]{l}t_{H}s_{H}=s_{H}t_{H}=t_{H}\\ t_{H}^{2}-(\alpha+\beta)t_{H}+\frac{\alpha\beta}{2}(1+s_{H})=0\end{array}\right.$$

$$\left\{\begin{array}[]{l}t_{s}s=st_{s}=t_{s}\\ t_{s}^{2}-(\alpha+\beta)t_{s}+\frac{\alpha\beta}{2}(1+s)=0\end{array}\right.$$

$$\omega =\sum _{H\in \mathcal{A}}\left({\lambda }_H+\sum _{Ker(s-1)=H}{\mu }_{s}s\right)f_H{\omega }_H\in {\Omega }^1(X)\otimes \text{mathbbm}CW⋉\text{mathbbm}C\mathcal{L}$$

$$s{f}_{H_0}{t}_Z=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}s{f}_{H_0}{\lambda }_H^{(i)}{s}_H^i{f}_H=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}{\lambda }_H^{(i)}s{s}_H^i{f}_{s_H^{-i}(H_0)}{f}_H$$

$$t_{Z}s{f}_{H_0}=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}{\lambda }_H^{(i)}{s}_H^i{f}_{H}s{f}_{H_0}=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}{\lambda }_H^{(i)}s(s^{-1}{s}_{H}s{)}^i{f}_{s^{-1}(H)}{f}_{H_0}$$

$$=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}{\lambda }_{s^{-1}(H)}^{(i)}s{s}_{s^{-1}(H)}^i{f}_{s^{-1}(H)}{f}_{H_0}=\sum _{\stackrel{H\supset Z}{0\le i<m_H}}{\lambda }_H^{(i)}s{s}_H^i{f}_H{f}_{H_0}$$

$$e_{H_1}{e}_{H_2}=\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s{f}_{H_s}\right)e_{H_2}=e_{H_1}\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s{f}_{H_s}\right)$$

$$\omega =\sum _{H\in \mathcal{A}}\left(\left({\lambda }_H+\sum _{Ker(s-1)=H}{\mu }_{s}s\right)-e_H\right)f_H{\omega }_H\in {\Omega }^1(X)\otimes B{r}^0(W,\mathcal{L})$$

$$[t_{H_0},t_Z]=[{\phi }_{H_0}{f}_{H_0}-e_{H_0},{\psi }_Z-\sum _H{e}_H]=-[{\phi }_{H_0}{f}_{H_0},\sum _H{e}_H]-[e_{H_0},{\psi }_Z-\sum _H{e}_H]$$

$$s{f}_{H_0}(\sum _H{e}_H)=f_{H_0}s(\sum _H{e}_H)=f_{H_0}(\sum _H{e}_H)s=(\sum _H{e}_H)s{f}_{H_0}$$

$$e_{H_0}\sum _H{e}_H=\sum _H{e}_{H_0}{e}_H=\sum _H{e}_{H_0}\sum _{s(H)=H_0}{\mu }_s{f}_{H_s}s=e_{H_0}\sum _{s\in \mathcal{R};\mathrm{Ker}(s-1)\supset Z}{\mu }_s{f}_{H_s}s=e_{H_0}\left({\psi }_Z-\sum _{H\supset Z}{\lambda }_H{f}_H\right)$$

$$[e_{H_0},{\psi }_Z-\sum _H{e}_H]=0$$

$$e_K{e}_{\underline{H}}=e_K{e}_L{e}_{\underline{J}}\in \text{mathbbm}Q[\delta ]W{e}_L{e}_{\underline{J}}\subset \text{mathbbm}Q[\delta ]W{e}_{\underline{H}}$$

$$|W|+|\mathcal{R}|\times |W|/2$$

$$e_{H_1}{e}_{H_2}=\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s\right)e_{H_2}=e_{H_1}\left(\sum _{s\in \mathcal{R}|s(H_2)=H_1}{\mu }_{s}s\right)$$

$$N_0=\{w\in W;w(H_0)=H_0\}=N_W(W_0)$$

$$\tilde{M}=\underset{H\in {\mathcal{A}}_0}{⨁}{V}_H$$

$$p_{H_0}(x)=\sum _{u(H)=H_0}{\mu }_{u}u.x$$

$$w{p}_0.(g_H\otimes y)=w\sum _{u(H)=H_0}{\mu }_{u}u.(g_H\otimes y)=\sum _{u(H)=H_0}{\mu }_{u}wu{g}_H\otimes y$$

$$p_{0}w.g_H\otimes y=p_0.(w{g}_H)\otimes y=\sum _{vw(H)=H_0}{\mu }_v(vw{g}_H)\otimes y=\sum _{w{v}^w(H)=H_0}{\mu }_v(w{v}^w{g}_H)\otimes y$$

$$=\sum _{v^w(H)=H_0}{\mu }_{v^w}w{v}^w{g}_H\otimes y$$

Full text

1. 1 Introduction

2. 1.1 Context

3. 1.2 Truncations

4. 1.3 Extensions and deformations

5. 1.4 Brauer algebras and Krammer representations

6. 2 The Brauer-Chen algebra and its extensions

7. 2.1 Presentations

8. 2.2 Chen’s additional relations

9. 2.3 Flat connection

10. 2.4 Finiteness of dimension

11. 2.5 A generalized Vogel algebra

12. 2.6 Lattice extensions

13. 3 The ideals $I_{r}$

14. 4 Connections with the Cohen-Frenk-Wales algebra

15. 5 Representations of $Br_{1}(W)$

16. 5.1 A direct presentation for $Br_{1}(W)$

17. 5.2 The $Br_{1}(W)$-modules of the form $\tilde{M}$ : definition.

18. 5.3 The $Br_{1}(W)$-modules of the form $\tilde{M}$ : properties.

19. 6 Proof of Theorem 1.1

20. 7 Examples

21. 7.1 Example : $G(4,2,2)$

22. 7.2 Example : $G(e,e,3)$

1. Introduction

1.1. Context

The algebra of Brauer diagrams was introduced by Brauer in [4] in order to understand the commutant of the orthogonal (or symplectic) groups of the $n$-fold tensor powers of a quadratic space $V$, in the same way as the (group algebra of the) symmetric group $\mathfrak{S}_{n}$ captures the commutant of the general linear group on the $n$-fold tensor powers of a vector space $V$. Its structure has been determined by Wenzl in [21]. Combinatorially, it can be described as an extension of the group algebra of $\mathfrak{S}_{n}$.

It has been shown that natural generalizations of the Brauer algebra exists for other finite reflection groups. In [8], Cohen Frenk and Wales attached one to any Coxeter group of type ADE, the case of type $A_{n}$ corresponding to the original case. A complete structural description was obtained in this case.

Later, Chen proposed in [6] a much larger generalization to arbitrary (complex) reflection groups. This generalization is isomorphic to the Cohen-Frenk-Wales algebra for real reflection groups of type ADE.

By contrast with the ADE case, the structural description of this algebra in general is still open, and even the determination of its dimension is still an open problem in general. In this paper we provide a partial description, and more precisely we decompose its first non-trivial quotient (’truncation’).

Another important aspect of the Brauer-Chen algebra is that it admits a natural deformation, obtained by the monodromy of some 1-form, in the same vein as the deformation of the group algebra known as the Hecke algebra. It has been proved recently that these Hecke algebras admit natural extensions. We then prove that the deformation of the Brauer-Chen algebra also admits such a natural extension, also provided by a monodromy construction.

Finally, we will explain how these two aspects should play a role in the construction and understanding of a generalized Krammer representation for the corresponding braid groups.

1.2. Truncations

We introduce a natural series of two-sided ideals $I_{1}\supset I_{2}\supset\dots$ and call representation of level $r$ of the Brauer-Chen algebra $Br(W)$ any representation factorizing through $Br_{r}(W)=Br(W)/I_{r+1}$ but not through $Br_{r-1}(W)$. This series of ideals satisfies that $Br_{0}(W)$ coincides with the group algebra of $W$, so $Br_{1}(W)$ is the first non-trivial quotient, and the first new representations are the ‘level 1 representations’ arising there.

Our first main result is then the following one.

Theorem 1.1**.**

Let ${\boldsymbol{\kappa}}$ be a field of characteristic [math], $\mathbf{k}={\boldsymbol{\kappa}}(\delta)$ the function field in one indeterminate. Let $W$ be (pseudo-)reflection group, with set of reflection hyperplanes $\mathcal{A}$, and $Br(W)$ the generic Brauer-Chen algebra defined over $\mathbf{k}$, as in section 2.1. Let us pick a representative $H_{0}$ for each $\mathcal{A}_{0}\in\mathcal{A}/W$. Then $Br_{1}(W)$ is a semisimple algebra, whose irreducible representations not factoring through $\overline{\mathbbm{Q}}W$ are in 1-1 correspondence with the pairs $(\mathcal{A}_{0},\theta)$ where $\mathcal{A}_{0}\in\mathcal{A}/W$ and $\theta$ is an irreducible representation of $N(W_{H_{0}})/W_{H_{0}}$ over $\mathbf{k}$. The restriction of such a representation to $W\subset Br_{1}(W)$ is the induced representation $\mathrm{Ind}_{N(W_{H_{0}})}^{W}\theta$.

The dimension of $Br_{1}(W)$ is equal to

[TABLE]

The above result provides a complete description of $Br(W)$ exacly when $Br(W)=Br_{1}(W)$, that is when there is no pair of transverse reflecting hyperplanes. This is the case for all irreducible rank 2 groups, but also for a few groups in rank 3. Specifically, this is the case for the groups $G(e,e,3)$ with $e\geq 3$ (see [20], Proposition 5.3.2) as well as for the primitive reflection groups $G_{24}$ and $G_{27}$ (but it is not the case for the Coxeter group $H_{3}$). Therefore, we get the following.

Corollary 1.2**.**

(See section 7.2) When $W=G(e,e,3)$ and $e$ is odd, $Br(W)$ is semisimple and has dimension $3e^{2}(2+3e)$.

In particular, this provides additional evidence towards conjecture 5.4.2 in [20] about the structure of a generalized BMW algebra in type $G(e,e,3)$.

A major obstacle to extend the above result to the whole algebra $Br(W)$ is that a good understanding in the general case is still missing of what was called in [8], for type ADE, admissible collections of transverse hyperplanes, and described there in terms of the associated root system.

1.3. Extensions and deformations

Our second main result (see theorem 2.6) is the following one. We proved in [17, 18] that Hecke algebras admit natural extensions by the Möbius algebra $\mathbbm{C}\mathcal{L}$ of the lattice $\mathcal{L}$ of the reflection subgroups of $W$, and that these algebras are monodromic deformation of $\mathbbm{C}W\ltimes\mathbbm{C}\mathcal{L}$ in the same way as the Hecke algebra is a monodromic deformation of $\mathbbm{C}W$. Here we prove that the same phenomenon occurs for the Brauer-Chen algebra. In particular, there is a natural KZ-type connection on $Br(W)\ltimes\mathbbm{C}\mathcal{L}$ that ‘covers’ in some sense these two different constructions, and which should be related, when $W=\mathfrak{S}_{n}$, with the tied-BWM algebra introduced by Aicardi and Juyumaya in [1]. In the framework of links invariants, this tied-BMW algebra supports the Markov trace responsible for the Kauffman and HOMFLY as well as their ‘tied’ variants.

Our work then suggests that there should exist a general definition of a generalized (tied-)BMW algebra for arbitrary complex reflection groups, that should be a module of finite type over its natural ring of definition. An idea of how much this module structure could be complicated for specific parameters is given e.g. by the extensive work of Yu on a BMW algebra for the family $G(d,1,n)$, see [22].

1.4. Brauer algebras and Krammer representations

The very first prototype of the representations of $Br_{1}(W)$ described here were implicitely described in [16], as they served as a guide to Chen to construct his algebra. These representations can be deformed using monodromy means to the generalized Krammer representations of the braid group of $W$ in type ADE as defined in [10, 12], see [15]. The explicit (algebraic, combinatorial) construction of these generalized Krammer representation for arbitrary complex reflection groups is quite an ongoing challenge. To my knowledge, the first successful attempt in this direction in the non-real case has been made by Neaime in [20], where such a matrix representation was constructed for $W=G(3,3,3)$, using a deformation of the Brauer-Chen algebra. It appears in this case that this representation admits non-trivial Galois conjugates. These conjugate representations are deformations of other level 1 representations of the Brauer-Chen representation, and this provides another motivation for the present work, namely to describe infinitesimally the Galois conjugates of the generalized Krammer representation that we introduced in [16]. Note that Neaime also constructed a conjectural matrix model for $G(4,4,3)$.

In the same way that understanding the general structure of the BMW algebra is useful to deal with the Krammer representation, the results of the present paper should thus provide similar services in the ongoing process of understanding and constructing these generalized Krammer representations.

Acknowledgements. I thank F. Digne and N. Matringe for useful discussions, G. Neaime for useful comments, and the anonymous referee for detailed remarks.

2. The Brauer-Chen algebra and its extensions

2.1. Presentations

Let $W$ be a complex (pseudo-)reflection group, $\mathcal{R}$ its set of reflections, and $\mathcal{A}$ the set of reflecting hyperplanes. Two hyperplanes $H_{1},H_{2}$ are called transverse if $H_{1}\neq H_{2}$ and $\{H\in\mathcal{A}\ |\ H\supset H_{1}\cap H_{2}\}=\{H_{1},H_{2}\}$. In this case we write H_{1}\mathrel{\mathchoice{\ooalign{\hfil\displaystyle|\hfil$\crcr$\displaystyle\cap}}{\ooalign{\hfil\textstyle|\hfil$\crcr$\textstyle\cap}}{\ooalign{\hfil\scriptstyle|\hfil$\crcr$\scriptstyle\cap}}{\ooalign{\hfil\scriptscriptstyle|\hfil$\crcr$\scriptscriptstyle\cap}}}H_{2}. A transverse collection of hyperplanes is a subset $\underline{H}=\{H_{1},\dots H_{r}\}$ of $\mathcal{A}$ with the property that every two hyperplanes in the collection are transverse. Let $\mathbf{k}$ be a commutative ring with $1$, and $\delta\in\mathbf{k}^{\times}$ an invertible scalar. We fix the choice of one scalar parameter $\mu_{s}\in\mathbf{k}$ per reflection $s$, subject to the condition $\mu_{s}=\mu_{t}$ as soon as $s$ and $t$ are conjugate. We denote by $\underline{\mu}$ the collection of such parameters.

By definition, the Brauer-Chen algebra $Br(W)=Br_{\underline{\mu}}(W)$ attached to $\underline{\mu}$ is defined by generators $w\in W$, $e_{H},H\in\mathcal{A}$ together with the relations of $W$, the semi-direct type relations $we_{H}=e_{w(H)}w$ and the following relations

1. (1)

$\forall H\in\mathcal{A}\ \ e_{H}^{2}=\delta e_{H}\ \ \&\ \ e_{H}s=se_{H}=e_{H}$ whenever $\mathrm{Ker}(s-1)=H$ 2. (2)

H_{1}\mathrel{\mathchoice{\ooalign{\hfil\displaystyle|\hfil$\crcr$\displaystyle\cap}}{\ooalign{\hfil\textstyle|\hfil$\crcr$\textstyle\cap}}{\ooalign{\hfil\scriptstyle|\hfil$\crcr$\scriptstyle\cap}}{\ooalign{\hfil\scriptscriptstyle|\hfil$\crcr$\scriptscriptstyle\cap}}}H_{2}\Rightarrow e_{H_{1}}e_{H_{2}}=e_{H_{2}}e_{H_{1}} 3. (3)

If $H_{1}$ and $H_{2}$ are distinct and not transverse, then

[TABLE]

Notice that the two equalities in relation (3) can be deduced one from the other using the semi-direct type relations.

Remark 2.1**.**

A natural attempt to a further generalization would be to consider relations of the form $e_{H}^{2}=\delta_{H}e_{H}$ for scalars $\delta_{H}$ depending on $H$. By the semi-direct type relations such a scalar should depend only on the orbit of $H$ under $W$. But then, up to rescaling the coefficients $\underline{\mu}$ we can assume all such $\delta_{H}$’s are the same, at least when all of them are nonzero. Moreover, if some of them are zero this provides quotients of the orginial algebra by relations of the form $e_{H}^{2}$ so this also fits inside the original framework.

2.2. Chen’s additional relations

In the relations above, we removed one relation from Chen’s original definition, the relation called (1)’ in [6]. This relation is that $we_{H}=e_{H}w=e_{H}$ if $w(H)=H$ and there exists $H_{1},H_{2}\in\mathcal{A}$ such that $\mathrm{Ker}(w-1)\cap H=H_{1}\cap H_{2}$ and $H_{1},H_{2}$ are not transverse.

It is claimed without proof in [6] that this condition is equivalent to the following one

[TABLE]

Actually, the preprint version of [6] on the arxiv (arXiv:1102.4389v1) is the version of (1)’ given here, and the claim that (1)’ and (1)” are equivalent can be found only in the published version. But on the other hand, in the published version of (1)’ the element $w$ is additionally assumed to be a reflection, which makes (1)’ not only equivalent, but formally equal to (1)”. So we have to interprete this addition to be a typo (corroborated by some examples in Chen’s paper).

The implication $(1)^{\prime}\Rightarrow(1)^{\prime\prime}$ is because, if $s\in\mathcal{R}$ is such that $s(H)=H$ and $H$ and $H_{s}=\mathrm{Ker}(s-1)$ are not transverse, then, setting $H_{1}=H_{s}$, $H_{2}=H$ and $w=s$, we get $\mathrm{Ker}(w-1)\cap H=H_{1}\cap H_{2}$ and by $(1)^{\prime}$ the conclusion. Conversely, if $w,H_{1},H_{2}$ are as in the assumption of $(1)^{\prime}$, then $w$ belongs to the parabolic subgroup fixing $H_{1}\cap H_{2}$. If this rank 2 parabolic subgroup is a dihedral group, then we have the conclusion because

1. (1)

either $w$ is a reflection, and we have the conclusion by (1)” 2. (2)

otherwise, setting $s$ the reflection w.r.t. $H$, we have that $ws$ is a reflection satisfying the same assumptions, and writing $w=ws.s$ we get the conclusion by applying (1)’ twice.

Therefore, (1)’ and (1)” are equivalent in a number of cases, including all Coxeter groups. It is however not true, in general that (1)” implies (1)’. In order to check this, we consider the complex reflection group $W$ of type $G(4,2,2)$ (see section 7.1 below), and apply to the given presentations a Gröbner basis algorithms, using the GAP4 package GBNP (see [11]), for a given value of $\delta\in\mathbf{k}=\mathbbm{Q}$. We obtain the dimensions $28$ and $40$ depending whether we add (1)’ or (1)”, and $64$ without both of them.

In [6] §9, it is argued that a reason for relation (1)’ to be added is that it may be closer to a previously introduced algebra in Coxeter type $B$ and more generally in type $G(d,1,n)$ (see [13]). Our purpose here being to consider the largest possible finite-dimensional algebra we consider it better to eliminate this. As noted by Chen, this condition is void in the usual (type $A$) case, so we get indeed a generalization of the usual Brauer algebra.

2.3. Flat connection

In any case, we reprove Proposition 5.1 of [6] in order to make it clear that these additional relations are not needed for the associated connection to be flat. Actually, we notice that our relation (1) is not needed either. Therefore, we let $Br^{0}(W)$ denote the algebra defined as $Br(W)$ but with relation (1) removed. We remark that this algebra is actually defined over $\mathbbm{Z}[\underline{\mu}]$ and has infinite rank. To every hyperplane $H$ we associate the logarithmic 1-form $\omega_{H}=(1/\pi\mathrm{i})\mathrm{d}\alpha_{H}/\alpha_{H}$ where $\alpha_{H}$ is an arbitrary linear form with kernel $H$. This 1-form is defined on the complement of the hyperplane arrangement $\mathcal{A}$, that we denote $X$.

Proposition 2.2**.**

(Chen) Assume $\mathbf{k}=\mathbbm{C}$. Then, the following 1-form

[TABLE]

is integrable and $W$-equivariant.

Proof.

We let $t_{H}=\left(\sum_{Ker(s-1)=H}\mu_{s}s\right)-e_{H}$. It is clear that $wt_{H}w^{-1}=t_{w(H)}$ hence we only need to prove that Kohno’s holonomy relations of [14] are satisfied. We recall these relations now. Let $Z$ be a codimension 2 flat, and $t_{Z}=\sum_{H\supset Z}t_{H}$. One needs to prove $[t_{Z},t_{H}]=0$ for all $H\supset Z$. If $Z$ is what Chen calls a crossing edge, that is if it is contained in exactly two hyperplanes $H_{1}$ and $H_{2}$, then it is clear that all the elements involved in $t_{H_{1}}$ and $t_{H_{2}}$ commute with each other, whence $[t_{H_{1}},t_{H_{2}}]=0$ hence $[t_{Z},t_{H}]=0$ for all $H\supset Z$.

If not, letting $\varphi_{H}=\sum_{Ker(s-1)=H}\mu_{s}s$ and $\varphi_{Z}=\sum_{H\supset Z}\varphi_{H}$, we first notice that $[\varphi_{H},\varphi_{Z}]=0$ from the integrability of the Cherednik connection (see e.g. [5]). Let us pick $H_{0}\supset Z$. We have

[TABLE]

But, for all $s\in\mathcal{R}$ with $\mathrm{Ker}(s-1)\supset Z$, we have $s(\sum_{H}e_{H})=(\sum_{H}e_{H})s$ hence $[\varphi_{H_{0}},\sum_{H}e_{H}]=0$. It remains to compute $[e_{H_{0}},\varphi_{Z}-\sum_{H}e_{H}]$. We have

[TABLE]

and similarly $(\sum_{H}e_{H})e_{H_{0}}=\varphi_{Z}e_{H_{0}}$ hence

[TABLE]

and this proves the claim.

∎

2.4. Finiteness of dimension

Chen proved that his algebra has finite rank as a $\mathbf{k}$-module. Actually, following the same lines of proof, one can prove the following more general statement.

Proposition 2.3**.**

Let $\mathbf{k}$ be a commutative ring with $1$, $\underline{Q}=(Q_{H})_{H\in\mathcal{A}}$ a family of polynomials in one indeterminate over $\mathbf{k}$ such that $H_{1}\sim H_{2}\Rightarrow Q_{H_{1}}=Q_{H_{2}}$. Then the quotient $Br^{\underline{Q}}(W)$ of $Br^{0}(W)$ by the relations $e_{H}Q_{H}(e_{H})=0$ has finite rank over $\mathbf{k}$.

Proof.

By the semidirect product relations it is clear that every element $Br^{0}(W)$ is a linear combinations of terms of the form $we_{H_{1}}\dots e_{H_{r}}$ for $w\in W$. If $m$ is the maximum of the degrees of the $Q_{H}$, we claim that we need no term with $r$ larger than $m\operatorname{\mathrm{rk}}(W)$. We argue by contradiction, and consider a term with $r$ minimal but greater than $m\operatorname{\mathrm{rk}}(W)$ which can not be rewritten using smaller $r$’s. First of all, for every $i<r$ we have that the $H_{i}$ and $H_{i+1}$ are either equal or transverse, by minimality of $r$ and relation (3). In particular, $e_{H_{i}}$ commutes with $e_{H_{i+1}}$, and actually (by induction) with every $e_{H_{j}}$ for $j>i$ for the same reason. Therefore we can assume that

[TABLE]

with $\{J_{1},\dots,J_{k}\}$ of cardinality $k$, and in particular a transverse collection of hyperplanes. By the polynomial relation on the $e_{H}$’s and the minimality of $r$, we have $u_{i}\leq m$ for all $i$, hence $r\leq m\times k$. Let us pick for each $i$ a nonzero vector $v_{i}$ in the orthogonal complement of $J_{i}$ (with respect to some unitary form preserved by $W$). For $i\neq j$ we have that $J_{i}$ and $J_{j}$ are transverse, and thus $v_{i}$ and $v_{j}$ are orthogonal. Therefore the $v_{i}$ form an orthogonal family of cardinality $k$, thus $k\leq\operatorname{\mathrm{rk}}(W)$ and $r\leq m\operatorname{\mathrm{rk}}(W)$, a contradiction. This proves the claim. ∎

When $\mathbf{k}$ is a field, it is clear that all finite-dimensional representations of $Br^{0}(W)$ factorize through $Br^{\underline{Q}}(W)$ for some $\underline{Q}$. Moreover, note that, picking one root of $Q_{H}$ for each conjugacy class of hyperplanes provide a surjective morphism from $Br^{\underline{Q}}(W)$ to $Br(W)$ with parameter(s) corresponding to the root(s). Actually, from the proof of the classification of the irreducible representations of $Br_{1}(W)$ given below, it will be clear that all the irreducible representations of $Br_{1}^{\underline{Q}}(W)$ (over an algebraically closed field) factor through one of them.

2.5. A generalized Vogel algebra

Finally, we prove here that, when $W$ is a 2-reflection group, the Brauer-Chen algebra appears as a quotient of an algebra $\mathfrak{Q}(W)$ depending on two scalar parameters $\alpha,\beta$, and defined by generators and relations as follows. Generators are $t_{H},H\in\mathcal{A}$, $w\in W$, and the relations are the relations of $W$ together with the $W$-invariance and holonomy relations

[TABLE]

where $Z$ runs among the codimension 2 flats, and

[TABLE]

for all $H\in\mathcal{A}$. This algebra is a generalization of an algebra introduced by P. Vogel in the framework of Vassiliev invariants, see [19].

Since $W$ is a 2-reflection group, in the definition of the Brauer-Chen algebra one can set $\lambda_{H}=\mu_{s_{H}}$ for $s_{H}$ the only reflection with hyperplane $H$. Let us set $t_{H}=\lambda_{H}(1+s_{H})-e_{H}\in Br(W)$. Then it is readily checked that $t_{H}s_{H}=t_{H}$, and $t_{H}^{2}=2\lambda_{H}^{2}(1+s_{H})+\delta e_{H}-4\lambda_{H}e_{H}=2\lambda_{H}^{2}(1+s_{H})+(\delta-4\lambda_{H})e_{H}$. Then $t_{H}^{2}+(\delta-4\lambda_{H})t_{H}=2\lambda_{H}^{2}(1+s_{H})+(\delta-4\lambda_{H})e_{H}+(\delta-4\lambda_{H})\lambda_{H}(1+s_{H})-(\delta-4\lambda_{H})e_{H}=(\delta-2\lambda_{H})\lambda_{H}(1+s_{H})$, hence $t_{H}^{2}-(\alpha+\beta)t_{H}+\frac{\alpha\beta}{2}(1+s_{H})=0$ with $\alpha=2\lambda_{H}-\delta$ and $\beta=2\lambda_{H}$.

This proves the following.

Proposition 2.4**.**

If $W$ is a 2-reflection group and $\mu_{s_{H}}=\lambda_{H}$, $\alpha=2\lambda_{H}-\delta$ and $\beta=2\lambda_{H}$, then there exists a surjective morphism $\mathfrak{Q}(W)\twoheadrightarrow Br(W)$ which is the identity on $W$ and maps $t_{H}\mapsto\lambda_{H}(1+s_{H})-e_{H}$.

An intriguing open question is whether these algebras $\mathfrak{Q}(W)$ are finite dimensional in general. It is conjectured to be the case when $W=\mathfrak{S}_{n}$, and known to be true for $n\leq 5$ by [19].

When $W$ is a finite Coxeter group with generating set $S$, another presentation of $\mathfrak{Q}(W)$ is easily seen to be given by generators $t_{H}=t_{s_{H}},H\in\mathcal{A}$, $s\in S$, together with the Coxeter relations, the holonomy relations, $st_{u}=t_{sus}s$ for $s\in S$ and $u\in\mathcal{R}$, and

[TABLE]

for all $s\in S$.

2.6. Lattice extensions

We operate a mixture of these ideas together with the ones of [17, 18]. Let $\mathcal{L}$ denote an admissible lattice in the sense of [18], that is a $W$-invariant sublattice of the lattice of all full reflection subgroups of $W$, where full means that if $s\in W$ is a reflection, then $W$ contains all the reflections fixing $\mathrm{Ker}(s-1)$, with the following properties of containing the cyclic (full) reflection subgroups and the trivial subgroups.

We consider its Möbius algebra $\mathbf{k}\mathcal{L}$, and denote $f_{L},L\in\mathcal{L}$ the natural basis element. Recall from [18] that full reflection subgroups can be naturally indexed by the collection of their reflection hyperplanes, and so in particular we denote $f_{H}\in\mathbf{k}\mathcal{L}$, for $H\in\mathcal{A}$, the natural basis element associated to the full reflection subgroup fixing $H$.

The following proposition was proved in [17] under the additional unnecessary assumption that $\mathcal{L}$ is the lattice of all parabolic subgroups. Here we provide the general proof.

Proposition 2.5**.**

Assume $\mathbf{k}=\mathbbm{C}$. Let us choose a collection of scalars $\lambda_{H},H\in\mathcal{A}$ such that $\lambda_{w(H)}=\lambda_{H}$ for all $w\in W$. Then the following 1-form

[TABLE]

is integrable and $W$-equivariant over the hyperplane complement $X$.

Proof.

Let us assume that we have picked a ‘distinguished’ reflection $s_{H}$ for each $H\in\mathcal{A}$ with the property that $ws_{H}w^{-1}=s_{w(H)}$ for all $w\in\mathcal{A}$ and $\langle s_{H}\rangle=W_{H}$. We denote $\lambda_{H}^{(0)}=\lambda_{H}$ and $\lambda_{H}^{(i)}=\mu_{s_{H}}^{i}$ for $i>0$ and $H\in\mathcal{A}$. We set $t_{H}=\sum_{0\leq i<m_{H}}\lambda_{H}^{(i)}s_{H}^{i}f_{H}$, with $m_{H}$ the order of $s_{H}$. Let $Z$ be a codimension 2 flat, and $t_{Z}=\sum_{H\supset Z}t_{H}$. One needs to prove $[t_{Z},t_{H_{0}}]=0$ for all $H\supset Z$. For this we only need to prove that $[sf_{H_{0}},t_{Z}]=0$ for $s\in\langle s_{H}\rangle$. We do this. We have

[TABLE]

and

[TABLE]

[TABLE]

and so we only need to check that $f_{H}f_{H_{0}}=f_{s_{H}^{-i}(H_{0})}f_{H}$. It is sufficient to prove this for the maximal admissible lattice $\mathcal{L}=\mathcal{L}_{\infty}$ of all full reflection subgroups. In this case, $f_{H}f_{H_{0}}=f_{G}$ for $G$ the smallest full reflection subgroup containing $\langle s_{H},s_{H_{0}}\rangle$. Since $\langle s_{s_{H}^{-i}(H_{0})},s_{H}\rangle=\langle s_{H}^{-i}s_{H_{0}}s_{H}^{i},s_{H}\rangle=\langle s_{H_{0}},s_{H}\rangle$ we indeed get $f_{H}f_{H_{0}}=f_{s_{H}^{-i}(H_{0})}f_{H}$, and this proves the claim, $W$-invariance being obvious.

∎

We now denote $Br^{0}(W,\mathcal{L})$ the algebra presented by generators $w\in W$, $e_{H},H\in\mathcal{A}$, $f_{L},L\in\mathcal{L}$, together with the relations

• •

$w=w_{1}w_{2}$ if $w=w_{1}w_{2}$ inside $W$

• •

$we_{H}=e_{w(H)}w$

• •

H_{1}\mathrel{\mathchoice{\ooalign{\hfil\displaystyle|\hfil$\crcr$\displaystyle\cap}}{\ooalign{\hfil\textstyle|\hfil$\crcr$\textstyle\cap}}{\ooalign{\hfil\scriptstyle|\hfil$\crcr$\scriptstyle\cap}}{\ooalign{\hfil\scriptscriptstyle|\hfil$\crcr$\scriptscriptstyle\cap}}}H_{2}\Rightarrow e_{H_{1}}e_{H_{2}}=e_{H_{2}}e_{H_{1}}

• •

If $H_{1}$ and $H_{2}$ are distinct and not transverse, then

[TABLE]

• •

$e_{H}f_{L}=f_{L}e_{H}$

• •

$e_{H}f_{H}=f_{H}e_{H}=e_{H}$

• •

$f_{L_{1}}f_{L_{2}}=f_{L_{1}\vee L_{2}}$

The following then provides an upgrading of both Propositions 2.2 and 2.5.

Theorem 2.6**.**

Assume $\mathbf{k}=\mathbbm{C}$. Let us choose a collection of scalars $\lambda_{H}\in\mathbbm{C}$, $H\in\mathcal{A}$ such that $\lambda_{w(H)}=\lambda_{H}$ for all $w\in W$. Then the following 1-form

[TABLE]

is integrable and $W$-equivariant.

Proof.

As in the proof of Proposition 2.2, we start by setting $\varphi_{H}=\lambda_{H}+\sum_{Ker(s-1)=H}\mu_{s}s$, $t_{H}=(\varphi_{H}-e_{H})f_{H}$. It is clear that $wt_{H}w^{-1}=t_{w(H)}$ hence we only need to prove that Kohno’s relations are satisfied. Let $Z$ be a codimension 2 flat, and $t_{Z}=\sum_{H\supset Z}t_{H}$. One needs to prove $[t_{Z},t_{H}]=0$ for all $H\supset Z$. If $Z$ is a crossing edge, that is it is contained in exactly two hyperplanes $H_{1}$ and $H_{2}$, then it is clear that all the elements involved in $t_{H_{1}}$ and $t_{H_{2}}$ commute with each other, whence $[t_{H_{1}},t_{H_{2}}]=0$ hence $[t_{Z},t_{H}]=0$ for all $H\supset Z$.

If not, letting $\psi_{Z}=\sum_{H\supset Z}\varphi_{H}f_{H}$, let us pick $H_{0}\supset Z$. We first notice that $[\varphi_{H}f_{H},\psi_{Z}]=0$ from Proposition 2.5.

We have

[TABLE]

But, for all $s\in\mathcal{R}$ with $H_{0}=\mathrm{Ker}(s-1)\supset Z$, we have

[TABLE]

hence $[\varphi_{H_{0}},\sum_{H}e_{H}]=0$. It remains to compute $[e_{H_{0}},\psi_{Z}-\sum_{H}e_{H}]$. We have

[TABLE]

and similarly $(\sum_{H}e_{H})e_{H_{0}}=(\psi_{Z}-\sum_{H\supset Z}\lambda_{H}f_{H})e_{H_{0}}$ hence

[TABLE]

and this proves the claim.

∎

Note that the quotient of the $Br^{0}(W,\mathcal{L})$ by the relations $e_{H}=0$ provides the semidirect product $\mathbf{k}W\ltimes\mathbf{k}\mathcal{L}$, while the quotient by the relations $f_{H}=1$ provides $Br^{0}(W)$. We can similarly introduce the algebras $Br^{\underline{Q}}(W,\mathcal{L})$ and in particular $Br(W,\mathcal{L})$ by imposing the relations $e_{H}^{2}=\delta e_{H}$ for $H\in\mathcal{A}$. By a straightforward adaptation of its proof, one gets the following analog of Proposition 2.3.

Proposition 2.7**.**

Let $\mathbf{k}$ be a commutative ring with $1$, $\underline{Q}=(Q_{H})_{H\in\mathcal{A}}$ a family of polynomials in one indeterminate over $\mathbf{k}$ such that $H_{1}\sim H_{2}\Rightarrow Q_{H_{1}}=Q_{H_{2}}$. Then the quotient $Br^{\underline{Q}}(W,\mathcal{L})$ of $Br^{0}(W,\mathcal{L})$ by the relations $e_{H}Q_{H}(e_{H})=0$ has finite rank over $\mathbf{k}$.

The existence of the flat connection of Theorem 2.6 raises the following question :

Question 2.8**.**

When $W=\mathfrak{S}_{n}$ and $\mathcal{L}$ is the lattice of all reflection subgroups of $W$, does this monodromy representation of the braid group over $Br(W,\mathcal{L})$ provide the braid group representations factoring through the tied-BMW algebra of Aicardi and Juyumaya (see [1]) ?

3. The ideals $I_{r}$

To any transverse collection $\underline{H}=\{H_{1},\dots,H_{r}\}$ we associate $e_{\underline{H}}=e_{H_{1}}\dots e_{H_{r}}\in Br(W)$. We denote $I_{r}$ for $r\geq 1$ the left ideal of $Br(W)$ generated by the $e_{\underline{H}}$ for $\underline{H}$ of cardinality $r$ (or equivalently, at least $r$). We first prove

Lemma 3.1**.**

Each $I_{r}$ is a two-sided ideal of $Br(W)$.

Proof.

Let $\underline{H}$ be a transverse collection of cardinality $r$. One needs to prove that $e_{\underline{H}}w\in I_{r}$ and $e_{\underline{H}}e_{K}\in I_{r}$ for all $w\in W$ and $K\in\mathcal{A}$. One readily checks that $e_{\underline{H}}w=e_{w^{-1}(\underline{H})}$ and that $w^{-1}(\underline{H})$ is again a transverse collection, so this proves $e_{\underline{H}}w\in I_{r}$ and more generally $e_{\underline{H}}\mathbbm{Q}[\delta]w\subset I_{r}$. We now consider $e_{\underline{H}}e_{K}$. If $K\in\underline{H}$ then clearly $e_{\underline{H}}e_{K}=\delta e_{\underline{H}}\in I_{r}$. Otherwise, if $K$ is transverse to every hyperplane in $\underline{H}$ we have $e_{\underline{H}}e_{K}=e_{\underline{H}\cup\{K\}}=e_{K}e_{\underline{H}}\in I_{r}$. If not, then $\underline{H}=\underline{J}\cup\{L\}$ with $L$ not transverse to $K$. But then $e_{L}e_{K}\in e_{L}.\mathbbm{Q}[\delta]W$ hence $e_{\underline{H}}e_{K}\subset e_{\underline{H}}\mathbbm{Q}[\delta]W\subset I_{r}$ as we already proved. This proves the claim. ∎

We denote $Br_{r}(W)=Br(W)/I_{r+1}$, so in particular $Br_{0}(W)=\mathbbm{Q}[\delta]W$.

For any transverse collection $\underline{H}$, let $W_{\underline{H}}$ denote the subgroup generated by all the reflections w.r.t. a hyperplane in $\underline{H}$. For $w\in W_{\underline{H}}$, we have $we_{\underline{H}}=e_{\underline{H}}$, hence for arbitrary $w\in W$ the element $we_{\underline{H}}$ depends only on the class of $w$ modulo $W_{\underline{H}}$. Therefore, $ce_{\underline{H}}$ is well-defined for $c\in W/W_{\underline{H}}$.

Proposition 3.2**.**

The collection of the $cW_{\underline{H}}$ for $\underline{H}$ a (possibly empty) transverse collection of hyperplanes and $c\in W/W_{\underline{H}}$ is a spanning set for $Br(W)$.

Proof.

By the above remark it is equivalent to say that the collection of the $we_{\underline{H}}$ for $w\in W$ and $\underline{H}$ a transverse collection form a spanning set. For this we need to prove that multiplying on the left such elements by $g\in W$ and $e_{K}$ for $K\in\mathcal{A}$ can be written as a linear combination of such elements. Since this is clear for $g\in W$, we look at $e_{K}we_{\underline{H}}=we_{w^{-1}(K)}e_{\underline{H}}$ and we finally need to prove that $e_{J}e_{\underline{H}}$ is equal to such a linear combination. The proof is then similar to the one of the previous lemma : if $K$ is transverse to all hyperplanes inside $\underline{H}$ then $e_{K}e_{\underline{H}}=e_{\underline{H}\cup\{K\}}$ and we are done, the case $K\in\mathcal{H}$ is also clear, and otherwise we can write $\underline{H}=\{L\}\cup\underline{J}$ with $L$ not transverse to $K$ and

[TABLE]

and we are done. ∎

As a corollary, when $W$ is a 2-reflection group admitting a single conjugacy class of reflections, then $Br_{1}(W)$ is spanned by a family of cardinality

[TABLE]

Now consider the case of a Coxeter group with generating set $S$. One could consider the ideal $J_{2}$ generated by the $e_{r}e_{s}$ for $r,s\in S$ being non-adjacent nodes of the Coxeter diagram. It is clear that $J_{2}\subset I_{2}$. Conversely, if $\underline{H}=(H_{1},H_{2})$ is a transverse collection, the parabolic subgroup $W_{Z}$ fixing $Z=H_{1}\cap H_{2}$ is generated by $s_{H_{1}}$ and $s_{H_{2}}$. It is conjugate to a standard parabolic $\langle r_{1},r_{2}\rangle$ with $r_{1},r_{2}\in S$, $s_{H_{i}}=wr_{i}w^{-1}$ for $i=\{1,2\}$ and some $w\in W$. Since $\mathrm{Ker}(r_{1}-1)$ and $\mathrm{Ker}(r_{2}-1)$ are also transverse we have that $r_{1}$ and $r_{2}$ are non-adjacent in the Coxeter graph hence $e_{r_{1}}e_{r_{2}}\in J_{2}$. But then $e_{H_{1}}e_{H_{2}}=we_{r_{1}}e_{r_{2}}w^{-1}\in J_{2}$ hence $I_{2}\subset J_{2}$ and this proves that $I_{2}=J_{2}$. This provides a sometimes more handy description of $Br_{1}(W)$ in the Coxeter case.

4. Connections with the Cohen-Frenk-Wales algebra

In [8], Cohen, Frenk and Wales associated a Brauer algebra to any Coxeter group of type ADE as follows. If $(W,S)$ is such a Coxeter system, it is defined by generators $r\in S$, $e_{r},r\in S$ and the Coxeter relations on $S$ together with the following ones, where the notation $r\sim s$ means that $r$ and $s$ are connected in the Coxeter diagram :

• •

(RSre) $re_{r}=e_{r}$

• •

(RSer) $e_{r}r=e_{r}$

• •

(HSee) $e_{r}^{2}=\delta e_{r}$

• •

(HCee) if $r\not\sim s$ then $e_{r}e_{s}=e_{s}e_{r}$

• •

(HCer) if $r\not\sim s$ then $e_{r}s=se_{r}$

• •

(RNrre) if $r\sim s$ then $rse_{r}=e_{s}e_{r}$

• •

(HNrer) if $r\sim s$ then $se_{r}s=re_{s}r$

We denote this algebra by $BrCFW(W)$. There is a natural injective map $W\to BrCFW(W)$ mapping $S$ to $S$ identically.

For every $t\in\mathcal{R}$ there is $s\in S$ and $w\in W$ such that $t=wsw^{-1}$.

We replace (RNrre) and (HNrer) by the following two conditions

• •

(RNrre’) if $r\sim s$ then $rsre_{s}=e_{r}e_{s}$

• •

(HNrer’) if $r\sim s$ then $rsre_{s}=e_{r}srs$

Under the other conditions, it is readily checked that (RNrre) is equivalent to (RNrre’) and that (HNrer) is equivalent to (HNrer’). From this it can be shown (see [6]) that this algebra is isomorphic to the Brauer-Chen algebra $Br(W)$.

The irreducible representations on which $I_{2}$ vanishes which are described in [8] are indexed by a $W$-orbit of positive roots – which can be identified to a $W$-orbit of reflecting hyperplanes – together with an irreducible character of a subgroup (called $W(C)$ in [8]) of $W$, which is shown ([8], Proposition 4.7) to be a complement of the parabolic subgroup $W_{0}$ fixing a given hyperplane $H_{0}$ inside the normalizer $N_{W}(W_{0})=\{w\in W\ |\ w(H_{0})=H_{0}\}$ of $W_{0}$, and is therefore isomorphic to $N(W_{0})/W_{0}$. This complement is described as the reflection subgroup generated by the reflections associated to the roots orthogonal to the highest one in the $W$-orbit under consideration. The construction of the representations is based on the root system. We show in the next section that these constructions can be made and generalized in a way independent of the chosen root system to arbitrary complex reflection groups.

5. Representations of $Br_{1}(W)$

In this section we denote ${\boldsymbol{\kappa}}$ a field of characteristic [math], and we assume $\mathbf{k}={\boldsymbol{\kappa}}(\delta)$ is the field of rational functions in $\delta$.

We will need the following easy lemma, for which we could not find a convenient reference.

Lemma 5.1**.**

The map $M\mapsto M\otimes_{{\boldsymbol{\kappa}}}\mathbf{k}$ induces a bijection between isomorphism classes of ${\boldsymbol{\kappa}}G$-modules and $\mathbf{k}G$-modules.

Proof.

Recall (from e.g. prop. 1 of [3], ch. V, annexe) the very general fact that $M_{1}\simeq M_{2}\Leftrightarrow M_{1}\otimes_{{\boldsymbol{\kappa}}}\mathbf{k}\simeq M_{2}\otimes_{{\boldsymbol{\kappa}}}\mathbf{k}$, so this map is injective. Since both ${{\boldsymbol{\kappa}}}G$ and $\mathbf{k}G$ are semisimple algebras over their base field, it is then sufficient to prove that every irreducible representation $\rho:G\to\mathrm{GL}_{n}(\mathbf{k})$ is isomorphic to the extension $\tilde{\rho}_{0}$ to $\mathbf{k}$ of a representation $\rho_{0}:G\to\mathrm{GL}_{n}({{\boldsymbol{\kappa}}})$. Since $G$ is finite and ${{\boldsymbol{\kappa}}}$ is infinite there exists $\delta_{0}\in{{\boldsymbol{\kappa}}}$ such that all entries of the $\rho(g)$, $g\in G$, viewed as rational functions, can be specialized at $\delta_{0}$ and such that $\delta_{0}$ is not a root of the $\det\rho(g)$, $g\in G$. Denote $\rho_{0}:G\to\mathrm{GL}_{n}({\boldsymbol{\kappa}})$ the corresponding specialization. Now notice that the values $\mathrm{tr}\rho(g)\in\mathbf{k}$ are algebraic over $\mathbbm{Q}\subset{{\boldsymbol{\kappa}}}$. But all the elements of $\mathbf{k}={{\boldsymbol{\kappa}}}(\delta)$ which are algebraic over ${{\boldsymbol{\kappa}}}$ actually belong to ${{\boldsymbol{\kappa}}}$, hence $\mathrm{tr}\rho(g)\in{\boldsymbol{\kappa}}$ for all $g\in G$. It follows that the character of $\rho$ is equal to the character of $\tilde{\rho}_{0}$ hence $\rho\simeq\tilde{\rho}_{0}$ and this proves the claim. ∎

5.1. A direct presentation for $Br_{1}(W)$

The algebra $Br_{1}(W)=Br(W)/I_{2}$ admits a more tractable presentation with the same generators, and for relations the relations of $W$, the semidirect type relations, and the following ones :

• •

$(1)$ $\forall H\in\mathcal{A}\ \ e_{H}^{2}=\delta e_{H}\ \ \&\ \ e_{H}s=se_{H}=e_{H}$ whenever $\mathrm{Ker}(s-1)=H$

• •

$(3_{1})$ If $H_{1}$ and $H_{2}$ are distinct, then

[TABLE]

In other terms, the defining relations $(2)$ and $(3)$ of $Br(W)$ together with the defining relations of $I_{2}$ are replaced by $(3_{1})$.

Indeed, relation $(2)$ modulo $I_{2}$ means H_{1}\mathrel{\mathchoice{\ooalign{\hfil\displaystyle|\hfil$\crcr$\displaystyle\cap}}{\ooalign{\hfil\textstyle|\hfil$\crcr$\textstyle\cap}}{\ooalign{\hfil\scriptstyle|\hfil$\crcr$\scriptstyle\cap}}{\ooalign{\hfil\scriptscriptstyle|\hfil$\crcr$\scriptscriptstyle\cap}}}H_{2}\Rightarrow e_{H_{1}}e_{H_{2}}=0, and this is equivalent to asserting $(3_{1})$ for $H_{1}$ and $H_{2}$, since there are no reflection mapping $H_{1}$ to $H_{2}$ when $H_{1}$ and $H_{2}$ are transverse (see [16], Lemma 3.1).

5.2. The $Br_{1}(W)$-modules of the form $\tilde{M}$ : definition.

Let $H_{0}\in\mathcal{A}$, $W_{0}=W_{\underline{H}_{0}}=\langle s_{0}\rangle$ be the pointwise stabilizer of $H_{0}$. We set

[TABLE]

the normalizer of $W_{0}$ in $W$. Let $M$ be a ${\boldsymbol{\kappa}}N_{0}$-module on which $W_{0}$ acts trivially. Then $\tilde{M}=\mathbf{k}W\otimes_{\mathbf{k}N_{0}}M_{\mathbf{k}}$ with $M_{\mathbf{k}}=\mathbf{k}\otimes_{{\boldsymbol{\kappa}}}M$ is a $\mathbf{k}W$-module which is the induced representation of $M_{\mathbf{k}}$. Let $\mathcal{A}_{0}=W.H_{0}\subset\mathcal{A}$ denote the orbit of $H_{0}$ under $W$. For all $H\in\mathcal{A}_{0}$ we pick $g_{H}\in W$ such that $g_{H}(H_{0})=H$. The set $\mathcal{G}_{0}$ of all such $g_{H}$ is a set of representatives of $W/N_{0}$, hence $\tilde{M}$ admits a direct sum decomposition

[TABLE]

with $V_{H}=g_{H}\otimes M_{\mathbf{k}}$ such that $wV_{H}\subset V_{w(H)}$ for all $w\in W$. Notice that $V_{0}=V_{H_{0}}$ is canonically identified to $M$ as an $N_{0}$-module, under the natural inclusion $N_{0}\subset W$.

We define $p_{0}=p_{H_{0}}\in\mathrm{End}(\tilde{M})$ by

[TABLE]

where the $u$’s are understood to be reflections (and the $\mu_{u}$’s are the defining parameters of the algebra associated to them), and $x\in V_{H}$, except when $H=H_{0}$ in which case $p_{H_{0}}(x)=\delta x$.

Lemma 5.2**.**

1. (1)

For all $w\in N_{0}$ and $x\in\tilde{M}$ we have $wp_{0}.x=p_{0}w.x$ 2. (2)

$p_{0}(\tilde{M})\subset V_{H_{0}}$** 3. (3)

$p_{0}^{2}=\delta p_{0}$**

Proof.

We first prove (1). For $x\in V_{H_{0}}$ this is immediate. For $H\in\mathcal{A}\setminus\{H_{0}\}$ we have, for $w\in N_{0}$ and $y\in M$,

[TABLE]

while

[TABLE]

[TABLE]

and this proves the claim, as $v\mapsto v^{w}$ is a bijection of $\mathcal{R}$. We now prove (2). For all $u$ with $u(H)=H_{0}$ we have $u.(g_{H}\otimes y)=ug_{H}\otimes y$ and $ug_{H}(H_{0})=u(H)=H_{0}$ hence $ug_{H}\in\mathbbm{C}N_{0}$ and this proves the claim. (3) is then an immediate consequence.

∎

Lemma 5.3**.**

For $w\in W$, and $H=w(H_{0})$, then $p_{H}\in\mathrm{End}(\tilde{M})$ defined by $p_{H}.x=wp_{0}w^{-1}.x$ depends only on $H$. If $H\in\mathcal{A}\setminus\mathcal{A}_{0}$ is not a conjugate of $H_{0}$ then we set $p_{H}=0$. If $H_{1},H_{2}\in\mathcal{A}$ satisfy $H_{2}=w(H_{1})$ for some $w\in W$, then $wp_{H_{1}}w^{-1}.x=p_{H_{2}}.x$ for all $x\in\tilde{M}$.

Proof.

Assume that $w_{1},w_{2}\in W$ satisfy $w_{1}(H_{0})=w_{2}(H_{0})$. Then $w=w_{2}^{-1}w_{1}\in N_{0}$ hence $wp_{0}w^{-1}.x=x$ for all $x\in\tilde{M}$ by the previous lemma (1) and this implies $w_{1}p_{0}w_{1}^{-1}.x=w_{2}p_{0}w_{2}^{-1}.x$ for all $x\in\tilde{M}$ and this proves the first part of the lemma. For the second part, if $H_{2}\not\in\mathcal{A}_{0}$ then $H_{1}\not\in\mathcal{A}_{0}$ too, hence $p_{H_{1}}=p_{H_{2}}=0$ satisfy the property. If not, let $w_{0}\in W$ such that $H_{1}=w_{0}(H_{0})$. Then $H_{2}=w(H_{1})=(ww_{0})(H_{0})$. It follows that, for all $x\in\tilde{M}$, we have $p_{H_{2}}.x=(ww_{0})p_{0}(ww_{0})^{-1}.x=w(w_{0}p_{0}w_{0}^{-1})w^{-1}.x=wp_{H_{1}}w^{-1}.x$ and this proves the claim. ∎

A consequence of the definition is that, for $w(H_{0})=H_{1}$, we have

[TABLE]

Since, on $V_{H_{2}}$, $p_{0}$ coincides with the action of $\sum_{u(H_{2})=H_{0}}\mu_{u}u$, this implies that

[TABLE]

for all $H_{2}\in\mathcal{A}$. Then, since $H=w(H_{0})$, we have

[TABLE]

which is equal by the above to

[TABLE]

[TABLE]

where as usual $u,v$ are assumed to belong to $\mathcal{R}$, and this proves

[TABLE]

for all $H_{1},H_{2}\in\mathcal{A}_{0}$.

We now want to prove that

[TABLE]

For this we note that

[TABLE]

and we conclude as before.

Finally, we need to check that $p_{0}s=sp_{0}=p_{0}$ whenever $\mathrm{Ker}(s-1)=H$. We have $p_{0}(\tilde{M})=V_{H_{0}}=1\otimes M$, and $s.(1\otimes y)=s\otimes y=1\otimes s.y$. But $W_{0}$ acts trivially on $M$ by assumption, hence $sp_{0}=p_{0}$. Then $p_{0}s=s(s^{-1}p_{0}s)=sp_{s^{-1}(H_{0})}=sp_{0}=p_{0}$, and through $W$-conjugation we get $p_{H}s=sp_{H}=p_{s}$ for all $H\in\mathcal{A}_{0}$, and $\mathrm{Ker}(s-1)=H$, the case $H\not\in\mathcal{A}_{0}$ being trivial.

This proves that $e_{H}\mapsto p_{H}$ extends the $\mathbf{k}W$-module structure on $\tilde{M}$ to a $Br_{1}(W)$-module structure.

5.3. The $Br_{1}(W)$-modules of the form $\tilde{M}$ : properties.

Proposition 5.4**.**

Let $M$ be a ${\boldsymbol{\kappa}}N_{0}$-module on which $W_{0}$ acts trivially. If $M$ is irreducible, then $\tilde{M}$ is an irreducible $Br_{1}(W)$-module. Moreover, if $M_{1}$ and $M_{2}$ are two such irreducible modules, then $\tilde{M}_{1}\simeq\tilde{M}_{2}$ iff $M_{1}\simeq M_{2}$.

Proof.

For the irreducibility, we adapt the arguments of [8], Proposition 5.3. Assume that $M$ is irreducible, and let $U\subset\tilde{M}$ a nonzero $Br_{1}(W)$-invariant subspace, and $q\in U\setminus\{0\}$.

We have $q=\sum_{H}\lambda_{H}g_{H}\otimes y_{H}$ for some $y_{H}\in M_{\mathbf{k}}$ and a collection of $\lambda_{H}\in\mathbf{k}={\boldsymbol{\kappa}}(\delta)$. Each $y_{H}\in M_{\mathbf{k}}$ can be written $y_{H}=\sum_{i}\nu_{H,i}m_{H,i}$ for some $m_{H,i}\in M$ and $\nu_{H,i}\in\mathbf{k}$. Up to chasing denominators, we can assume $\lambda_{H},\nu_{H,i}\in{\boldsymbol{\kappa}}[\delta]$. Let us pick $H_{1}$ and $i_{0}$ with $\lambda_{H_{1}}\nu_{H,i_{0}}$ being of maximal degree in $\delta$ among the terms $\lambda_{H}\nu_{H,i}\in{\boldsymbol{\kappa}}[\delta]$ with $y_{H}\neq 0$. Then $e_{H_{1}}.q=\sum_{H}\sum_{i}\lambda^{\prime}_{H}\nu^{\prime}_{H,i}g_{H_{1}}\otimes m^{\prime}_{H,i}$ for some $m^{\prime}_{H,i}\in M$ and $\lambda^{\prime}_{H},\nu^{\prime}_{H,i}\in{\boldsymbol{\kappa}}[\delta]$ with $\deg\lambda^{\prime}_{H}\nu^{\prime}_{H,i}\leq\deg\lambda_{H}\nu_{H,i}\leq\lambda_{H_{1}}\nu_{H_{1},i}$ whenever $H\neq H_{1}$. On the other hand, we have $\lambda^{\prime}_{H_{1}}\nu^{\prime}_{H_{1},i}=\delta\lambda_{H_{1}}\nu_{H_{1},i}$ and $m^{\prime}_{H_{1},i}=m_{H_{1},i}$ for all $i$. It follows that $e_{H_{1}}.q$ can be written as

[TABLE]

with the $m^{\prime\prime}_{j}\in M$, $\omega_{j}\in{\boldsymbol{\kappa}}[\delta]$ with degree (strictly) lower than maximal degree of the $\lambda_{H_{1}}\nu_{H_{1},i}$. Since $y_{H_{1}}\neq 0$, this implies that $e_{H_{1}}.q\neq 0$. On the other hand, $e_{H_{1}}.q\in U\cap V_{H_{1}}$ hence $U\cap V_{H_{1}}\neq\{0\}$. We can thus assume $q=g_{H_{1}}\otimes y\in U\cap V_{H_{1}}\setminus\{0\}$.

Since $M$ is irreducible, by Lemma 5.1 we have that $M_{\mathbf{k}}$ is irreducible as a $\mathbf{k}N_{0}$-module. It follows that $V_{H_{1}}\subset U$. Since $W$ acts transitively on the $V_{H}$, $H\in\mathcal{A}_{0}$, it follows that $U$ contains all the $V_{H}$ hence $U=\tilde{M}$ and $\tilde{M}$ is an irreducible $Br_{1}(W)$-module.

Now assume that a given $Br_{1}(W)$-module $\tilde{M}$ that is obtained from two simple ${\boldsymbol{\kappa}}N_{0}$-modules with the property that the action of $W_{0}$ is trivial. We thus have two a priori distinct direct sum decompositions

[TABLE]

with the property that the action of $N_{0}$ on $V_{H_{0}}^{(i)}$ is (isomorphic to) $M_{i}$ for $i\in\{1,2\}$. But since $V_{H_{0}}$ is the image of $p_{H_{0}}$ by Lemma 5.2 it follows that $(M_{1})_{\mathbf{k}}\simeq(M_{2})_{\mathbf{k}}$ as $\mathbf{k}N_{0}$-modules. But this implies that $M_{1}\simeq M_{2}$ as ${\boldsymbol{\kappa}}N_{0}$-modules and this proves the second part of the statement. ∎

Remark 5.5**.**

For the second part of the argument, we are not replicating the argument of [8], Proposition 5.4, because it looks incorrect to us. Indeed, it is claimed there that the restriction of $\tilde{M}$ to $W(C_{\mathcal{B}})$, which is equivalent to the restriction to our $N_{0}$, is isomorphic to $|\mathcal{R}|M$, while it is actually isomorphic to $\mathrm{Res}_{N_{0}}\mathrm{Ind}_{N_{0}}^{W}M$. Actually, already for $D_{4}$ one can find non-isomorphic modules $M_{1},M_{2}$ for which the restriction of $\tilde{M}_{1}$ and $\tilde{M}_{2}$ to $W$ are isomorphic.

Proposition 5.6**.**

Every irreducible representation of $Br_{1}(W)$ has the form $\tilde{M}$ for some irreducible $M$.

Proof.

Let $Q$ be an irreducible representation of $Br_{1}(W)$ not factoring through $\mathbf{k}W$. This means that there exists some $H_{0}\in\mathcal{A}$ for which $e_{H_{0}}$ acts nontrivially. Denote $e_{0}=e_{H_{0}}$, and $\mathcal{A}_{0}$ the orbit of $H_{0}$ under $W$. Since $e_{0}^{2}=\delta e_{0}$ and $e_{0}$ acts non trivially, there exists $x_{0}\in Q$ such that $e_{0}.x_{0}=\delta x_{0}$. Let $W_{0}$ denote the subgroup fixing $H_{0}$ and $N_{0}$ its normalizer. For $s\in W_{0}$ we have $s.x_{0}=(1/\delta)se_{0}.x_{0}=(1/\delta)e_{0}.x_{0}=(\delta/\delta)x_{0}=x_{0}$ hence $W_{0}$ acts trivially on $x_{0}$. Now we set $V=\mathbf{k}N_{0}.x_{0}\subset Q$. Note that $V$ is a $\mathbf{k}N_{0}$-module factoring through $\mathbf{k}N_{0}/W_{0}$. Moreover, for all $w\in N_{0}$, we have $e_{0}w.x_{0}=we_{0}x_{0}=\delta w.x_{0}$ hence $e_{0}$ acts as $\delta.\mathrm{Id}$ on $V$.

Let $H\in\mathcal{A}_{0}$, and $w\in W$ such that $H=w(H_{0})$. Then $w.V$ depends only on $H$. Indeed, if $w_{1}^{-1}w_{2}\in N_{0}$ we have $w_{1}^{-1}w_{2}V=V$ hence $w_{2}V=w_{1}V$. We set $V_{H}=w.V$. For $g\in W$ we have $g.V_{H}=gw.V=V_{H^{\prime}}$ for $H^{\prime}=gw(H)$. Clearly $\sum_{H}V_{H}$ is a $\mathbf{k}W$-submodule of $Q$. Then, for $x\in V_{H}$ and $H=w(H_{0})$, we have $e_{H}.x=we_{0}w^{-1}.x=we_{0}.(w^{-1}.x)=\delta w.(w^{-1}.x)=\delta.x$. But this implies that, either $e_{0}.x=(1/\delta)e_{0}e_{H}.x=0$ or

[TABLE]

hence $e_{0}$ maps $x$ inside $V$. In particular $\sum V_{H}$ is stable under $e_{0}$. Finally, if $H^{\prime}\in\mathcal{A}\setminus\mathcal{A}_{0}$, we have $e_{H^{\prime}}.x=(1/\delta)e_{H^{\prime}}e_{0}.x=0$ for all $x\in V$. This implies similarly that $e_{H^{\prime}}.wx=w.e_{w^{-1}(H^{\prime})}.x=0$ for all $x\in V$, hence $e_{H^{\prime}}$ acts by [math] on $\sum_{H}V_{H}$. This implies that $\sum_{H}V_{H}$ is stable under $Br_{1}(W)$, hence $\sum_{H}V_{H}=Q$.

Let now assume that $V$ is not irreducible as a $\mathbf{k}N_{0}$-module, and contains a proper irreducible submodule $U$. Then one proves similarly that $\sum_{H}U_{H}$ is stable under $Br_{1}(W)$, where $U_{H}=w.U$ for $w(H_{0})=H$, hence $\sum_{H}U_{H}=\sum_{H}V_{H}=Q$. Finally, remark that, if $x\in\sum_{H}V_{H}$ satisfies $e_{0}.x=\delta.x$, then $x\in V$, since we know that $e_{0}.(\sum_{H}V_{H})\subset V$. Similarly, we check that $e_{0}$ maps $\sum_{H}U_{H}$ to $U$, hence $U$ and $V$ can be both identified with the nullspace of $e_{0}-\delta$ on the same space, hence $U=V$, which proves the irreducibility of $V$ as a $\mathbf{k}N_{0}$-module.

Finally, for $H=w(H_{0})$ we have that $e_{H}=we_{0}w^{-1}$ maps $Q$ to $V_{H}=w.V$ and acts by $\delta$ on $V_{H}$. It follows that $V_{H}$ is the nullspace of $e_{H}-\delta$ and this proves that $Q=\bigoplus_{H}V_{H}$. This proves that, as a $\mathbf{k}W$-module, $Q$ is the induced representation of $V$. But, by Lemma 5.1, for any ${\boldsymbol{\kappa}}(\delta)N_{0}$-module $V$ there exists a ${\boldsymbol{\kappa}}N_{0}$-module $V_{0}$ such that $V\simeq V_{0}\otimes_{{\boldsymbol{\kappa}}}\mathbf{k}$, hence $Q\simeq\tilde{V}_{0}$, and this proves the claim.

∎

Remark 5.7**.**

The determination of the representations of $Br_{1}(W)$ for type $H_{3}$ was essentially done in [6].

6. Proof of Theorem 1.1

From Proposition 3.2 we know that $Br_{1}(W)$ admits a spanning set formed by the $w\in W$ and the $we_{H},H\in\mathcal{H}$, where $w\in W/W_{H}$ and $W_{H}$ is the parabolic subgroup fixing $H$. Therefore the dimension of $Br_{1}(W)$ is at most $|W|+\sum_{\mathcal{A}_{0}\in\mathcal{A}/W}|\mathcal{A}_{0}|\times|W|/|W_{H_{0}}|$ where $W_{H_{0}}$ is the pointwise stabilizer of a representative $H_{0}\in\mathcal{A}_{0}$.

On the other hand, from the previous section we know that $Br_{1}(W)$ has irreducible representations of dimension $|\mathcal{A}_{0}|\times\dim\theta$ where $\theta$ is an irreducible representation of $N_{W}(W_{H_{0}})/W_{H_{0}}$. Therefore its dimension is at least

[TABLE]

But since $W/N_{W}(W_{H_{0}})$ is in bijection with $|\mathcal{A}_{0}|$, this is equal to the previous quantity. Therefore we get the formula for the dimension given by the theorem, and semisimplicity as well.

7. Examples

7.1. Example : $G(4,2,2)$

We consider the group $W=G(4,2,2)$ of rank 2, made of monomial matrices with entries in $\mu_{4}=\{1,-1,\mathrm{i},-\mathrm{i}\}$ and whose product of the nonzero entries belongs to $\mu_{2}=\{-1,1\}$. It admits the presentation

[TABLE]

and therefore an automorphism of order 3 mapping $s\mapsto t\mapsto u\mapsto s$. It has order $16$, and 6 reflections, $s,t,u,s^{\prime},t^{\prime},u^{\prime}$, forming 3 conjugacy classes of two elements, $\{s,s^{\prime}\},\{t,t^{\prime}\},\{u,u^{\prime}\}$. The reflection $s^{\prime}$ is equal to $tst=t\underline{stu}u=t\underline{tus}u=usu$, hence $t^{\prime}=utu=sts$, $u^{\prime}=sus=tut$.

The normaliser $N_{0}$ of $W_{0}=\langle s\rangle$ is abelian of order $8$, isomorphic to $\mathbbm{Z}/2\times\mathbbm{Z}/4$. The element $z=stu$ generates the center of $W$, has order $4$, and its image generates $N_{0}/W_{0}$.

Therefore $N_{0}/W_{0}$ is naturally identified with $Z(W)$.

The irreducible representations of $N_{0}$ satisfying the property that $W_{0}$ acts trivially are therefore uniquely determined by a choice of $\zeta\in\mu_{4}$ and given by the formula $R_{\zeta}:z\mapsto\zeta$. We now consider the induced module $V_{s}(\zeta)=\mathrm{Ind}_{N_{0}}^{W}R_{\zeta}$. It admits for basis $v_{s},v_{s^{\prime}}=t.v_{s}$, where $N_{0}$ acts on $\mathbbm{C}v_{s}$ by $R_{\zeta}$. From this we get $u.v_{s}=ts.stu.v_{s}=\zeta ts.v_{s}=\zeta t.v_{s}=\zeta v_{s^{\prime}}$, hence $u.v_{s^{\prime}}=u^{-1}.v_{s^{\prime}}=\zeta^{-1}.v_{s}$, and similarly $s.v_{s}=v_{s}$, $s.v_{s^{\prime}}=u.ust.v_{s}=\zeta.u.v_{s}=\zeta^{2}.v_{s}$, and $t.v_{s}=v_{s^{\prime}}$, $t.v_{s^{\prime}}=v_{s}$. Therefore we get the matrices

[TABLE]

Notice that, for $\zeta\in\{-\mathrm{i},\mathrm{i}\}$, we get reflection representations for $W$. We have $s,s^{\prime}\in N_{0}$, while $xsx=s^{\prime}$ for $x\in\{t,u,t^{\prime},u^{\prime}\}$. Letting $s_{0}=s$, we get $p_{s}.v_{s}=\delta v_{s}$ and

[TABLE]

Since $s^{\prime}=tst$ we get

[TABLE]

and $p_{x}=0$ for $x\in\{t,t^{\prime},u,u^{\prime}\}$. Note that $1+\zeta+\zeta^{2}+\zeta^{3}=0$ unless $\zeta=1$.

Condition (1)” is always fulfilled on such representations when $p_{H}=0$, so we consider only the case where $H$ is the reflecting hyperplane of $s$ or $s^{\prime}$. But in this case the reflection under consideration has to be the other one, and therefore we need to check whether $sp_{s^{\prime}}=p_{s^{\prime}}s=p_{s^{\prime}}$ and $s^{\prime}p_{s}=p_{s}s^{\prime}=p_{s}$. This is the case if and only if $\zeta^{2}=1$, that is $\zeta\in\{-1,1\}$.

In order that condition (1)’ is fulfilled, we need first of all that (1)” is fulfilled, so we consider only the case $\zeta^{2}=1$. Let $w\in W$ satisfying $\,{}^{w}s=s$, that is $w\in N_{0}$. Since $Z(W)$ is a complement to $W_{0}$ inside $W$, and since it is generated by $z$, the condition is then whether $zp_{s}=p_{s}z=p_{s}$, which would imply $zp_{s^{\prime}}=p_{s^{\prime}}z=p_{s^{\prime}}$ after conjugation by $t$. But $zp_{s}=p_{s}z=\zeta p_{s}$, hence the condition is fulfilled only if $\zeta=1$.

These two facts prove that conditions (1)’ and (1)” are not equivalent, and that they are genuine additional conditions.

The other representations $V_{t}(\zeta)$, $V_{u}(\zeta)$ are deduced from $V_{s}(\zeta)$ by applying $\varphi$, as this is readily extended to an automorphism of order $3$ of $Br(M)$.

7.2. Example : $G(e,e,3)$

For $W=G(e,e,3)$, there is a single conjugacy class of reflections. Let us choose $s_{0}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix}$. Then $N_{0}$ is easily determined to be equal the image of $\langle\mu_{e},(1,2)\rangle<\mathrm{GL}_{2}(\mathbbm{C})$ under the map $M\mapsto\mathrm{diag}(M,q(M)^{-1})$ where $q(M)$ is the product of the entries of $M$. Therefore $N_{0}/W_{0}$ is isomorphic to the group of all complex $e$-th roots of $1$. Now, $W$ has order $e^{2}\times 3!=6e^{2}$, $N_{0}$ has order $2e$, $\mathcal{R}$ has cardinality $3e$, hence $Br(M)=Br_{1}(M)$ has dimension $6e^{2}+9e^{3}=3e^{2}(2+3e)$.