The Representations of the Brauer-Chen Algebra
Ilias Andreou




Research supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the Basic Research Financing (Horizontal support for all Sciences), National Recovery and Resilience Plan (Greece 2.0), Project Number: 15659, Project Acronym: SYMATRAL


The representations of the Brauer-Chen algebra
    Ilias Andreou
    
==============================================




In this paper, we determine the structure and representation theory of the Brauer algebra associated to a complex reflection group (here called the Brauer-Chen algebra), defined by Chen in 2011. We prove that it is semisimple and provide a construction for its simple modules for generic values of the parameters, in a uniform way for all complex reflection groups. We then apply these results to the cases of all irreducible complex reflection groups: for the groups in the infinite series, we obtain a numerical formula for the dimension of the corresponding Brauer algebra, and for all exceptional complex reflection groups, we compute the dimension of the corresponding Brauer algebra explicitly, using computational methods. We also obtain a uniformly defined basis for the Brauer algebra of any complex reflection group, defined over a field. Finally, we determine for which complex reflection groups the corresponding Brauer algebra is a free module over its ring of definition. 




§ INTRODUCTION

 



 §.§ Some history and motivation

In 1937, Richard Brauer defined a family of algebras, now named after him, which described the centralizer of the action of the orthogonal group on a tensor power of its natural representation <cit.>. For the corresponding action of the general linear group, the centralizer had earlier been identified as the group algebra of the symmetric group on the number of factors of the tensor representation, acting by permuting these factors. Such centralizer relations are often referred to as Schur-Weyl dualities.



The representation theory of Brauer algebras was determined in 1988 by Wenzl <cit.>. The techniques employed for this work were inspired by the famous earlier work of Jones connecting certain trace functions on Von Neumann algebras to invariants of links <cit.>. At the same time, Birman and Wenzl <cit.> and, independently, Murakami <cit.>, also inspired by Jones's work, introduced the - now also famous - BMW algebra. This is also an algebra which is closely related to link invariants, by supporting, notably, a trace that lifts to the Kauffman polynomial. The BMW algebra is a deformation of the Brauer algebra, the latter being isomorphic to special cases of the former, i.e., for certain specializations of the parameters.
In fact, it is a deformation of the Brauer algebra in much the same way as the Hecke algebra is a deformation of the group algebra of the symmetric group.  
 
In a spirit similar to how Hecke algebras have been defined and extensively studied for all complex reflection groups (yielding numerous connections to topics in representation theory, algebraic geometry, low-dimensional topology and even theoretical physics), there have been many efforts to extend the definition of Brauer and BMW algebras to complex reflection groups other than the symmetric group (which corresponds to the original case). We mention below some notable examples of these efforts.


In 2001, Häring-Oldenburg <cit.> introduced the cyclotomic BMW and Brauer algebras, i.e., BMW and Brauer algebras associated to complex reflection groups of type G(m,1,n), providing connections of the former to link invariants in the solid torus. In 2014, Bowman and Cox <cit.> associated a certain subalgebra of the cyclotomic Brauer algebra with the subgroups G(m,p,n) of G(m,1,n), which comprise almost all irreducible complex reflection groups.

In 2003, Cohen, Gijsbers and Wales <cit.> extended BMW algebras to finite Coxeter groups of simply laced type (that is, types A, D and E, with A being the original case of the symmetric group), which led, in a new way, to earlier discovered faithful representations of the Artin group of the corresponding Coxeter types. Motivated by this, in 2007, Cohen, Frenk and Wales <cit.> introduced Brauer algebras of type ADE, and completeley determined their representations for generic values of their parameters. Finally, extending this work, Cohen, Liu and Yu <cit.>, and Cohen-Liu <cit.> defined Brauer algebras of Coxeter types B and C, respectively.

The most uniform definition of a Brauer algebra for every complex reflection group was given by Z.Chen <cit.>, in 2011; we call this the Brauer-Chen algebra, as in <cit.>. It was at least in part intended to serve as a stepping stone for the definition of a general BMW algebra for all complex reflection groups. In particular, it is defined in a way as to support certain formal flat connections which would dictate its deformation to a general BMW algebra. Such a relation exists between complex reflection groups and the corresponding Hecke algebras <cit.>, and was shown by Marin <cit.> and, later, Chen <cit.>, to exist between the Brauer and BMW algebras of types A and ADE, respectively. As shown originally by Chen, the Brauer-Chen algebra associated to Coxeter groups of type ADE is isomorphic to the corresponding Brauer algebra of Cohen, Frenk, Wales, and, in type G(m,1,n), it contains the cyclotomic Brauer algebra of Häring-Oldenburg as a direct component. As stated by Cohen and Liu in the introduction of <cit.>, the Brauer algebras of type B and C are not isomorphic to the Brauer-Chen algebra of these types, but are connected in a way necessitating further research. 

Finally, we mention two more recent generalizations of BMW and Brauer algebras. In 2017, Chen <cit.> defined BMW algebras associated to every Coxeter system that give rise to certain irreducible representations of the corresponding Artin group. He conjectured that one can obtain these BMW algebras as deformations of the corresponding Brauer-Chen algebras of the same type, via a flat connection supported by the latter, as the ones mentioned above. To our knowledge this question is still unanswered.

In 2018, in his PhD thesis <cit.>, G. Neaime defined BMW and Brauer algebras of type G(m,m,n), which are the non-real equivalent of simply laced type, and obtained irreducible representations of the corresponding braid groups.  Again, apart from the case n=3 and m odd, for which he showed that his Brauer algebra is isomorphic to the Brauer-Chen algebra, the connections between the two algebras are unexplored.




 §.§ Overview of our results


Motivated by the uniform way of its definition, its relation to other defined Brauer algebras and its possible applications to the generalization of BMW algebras this article is devoted to the study of the structure of the generic Brauer-Chen algebra for any complex reflection group. 

In particular, we show that, for generic values of the parameters and suitable fields of definition (e.g. algebraically closed fields of characteristic zero, see Definition <ref>), the Brauer-Chen algebra is semisimple and we provide a complete determination of its simple modules, as well as a formula for its dimension (Theorems <ref> & <ref>, respectively). Such results were obtained for the Brauer algebra of simply laced type (ADE) by Cohen, Frenk and Wales in their original paper. They constructed the simple modules of the corresponding Brauer algebra from pairs consisting of a suitable collection of pairwise orthogonal roots (called admissible) and a simple module of a certain subgroup of the corresponding Coxeter group, depending on the chosen collection of orthogonal roots. Generalizing this construction was not straightforward since many of its elements (e.g. the admissibility condition) depended on the language of roots, which does not behave, for general complex reflection groups, as well as for Coxeter groups.
On the other hand, in <cit.> Marin constructed a certain class of simple modules for the Brauer-Chen algebra of every complex reflection group. In the cases of Coxeter groups of simply laced type, these modules correspond to the modules constructed by Cohen, Frenk and Wales arising from collections consisting of a single root (these are always admissible).


The key element that enabled the generalization of the above results is the observation that a simple module of the Brauer-Chen algebra is determined by a certain module of the stabilizer of a certain collection (many equivalent ones, in fact) of reflecting hyperplanes. In particular, as a module of the corresponding complex reflection group, it is induced by this module of the stabilizer. Given such a structure, one can reverse the procedure and study the conditions under which a collection of reflecting hyperplanes and a module of its stabilizer induce indeed a simple module of the Brauer-Chen algebra. This generalizes the notion of admissibility of Cohen, Frenk and Wales. After constructing all simple modules, it is by computing their dimensions and finding a suitably small spanning set that we obtain semisimplicity and the dimension of the algebra. What is noteworthy is that up to this point the treatment is uniform for all complex reflection groups, i.e. without having to resort to a case-by-case analysis, a standard technique when dealing with complex reflection groups.

Applying these results to the case of irreducible complex reflection groups of the infinite series G(m,p,n) suggests a more explicit structure of simple modules, closer to the constructions of Cohen, Frenk, Wales and Marin. In particular, for admissible collections of reflecting hyperplanes, the suitable modules that induce simple modules of the Brauer-Chen algebra are those that induce the trivial representation on a certain subgroup of the stabilizer of the given collection. This condition is then verified to hold for all exceptional complex reflection groups, using programming in GAP4 <cit.>, after a small modification (namely, a certain non-trivial representation replaces the trivial one on a certain subgroup of the stabilizer). This result provides a much more efficient formula for computing the dimension of the Brauer-Chen algebra, as well as a nice basis over the fields in question (Theorem <ref>).

The obtained basis specializes to the original basis of the Brauer algebra, as well as to the basis obtained by Cohen, Frenk and Wales for the Brauer algebra of simply laced types. In these situations it is, in fact, a basis over the ring of definition of the algebra. Generalizing this, we show that our obtained basis over a field is a basis over the ring of definition of the Brauer-Chen algebra for all irreducible complex reflection groups other than the exceptional groups G_25,G_32 (Propositions <ref>, <ref>, <ref>). For these last two groups, the Brauer-Chen algebra is not a free module over its ring of definition, which we show by noticing that for certain values of its parameters, the dimension of the algebra over the corresponding field changes (Proposition <ref>). Again, for the study of freeness of the Brauer-Chen algebra in the exceptional groups, we used programming in GAP4 to obtain the corresponding results.












§ PRELIMINARIES





 §.§ Basics of Complex Reflection Groups



Although our definitions here may be slightly modified, we refer the reader to the book <cit.> of Lehrer and Taylor for more detailed information on reflections and complex reflection groups. 

 Let V be a finite dimensional complex vector space. An element r∈(V) is called a pseudo-reflection if the subspace (r-1) of fixed vectors of r is of codimension 1 in V. The subspace (r-1) is called the reflecting hyperplane  of r and we denote it by H_r.



Here, we will just use the term reflection for any pseudo-reflection; over ℝ the notions coincide.  If r is a reflection of V, and l is an invertible linear transformation of V, then lrl^-1 is again a reflection, and H_lrl^-1=l(H_r).



 A complex reflection group W is a finite subgroup of (V_W) for some complex vector space V_W which is generated by reflections. The representation V_W is called the natural representation of W. If it is irreducible, then W is said to be irreducible. Finally, the rank of W is the dimension of V_W minus the dimension of the subspace of fixed vectors of W. 



For a complex reflection group W⊆(V_W), one can always assume that there is a positive definite hermitian form on V_W which is stable under W (take any positive definite hermitian form on V_W and normalize it under the finite group W). A reflection r of V_W respecting that form is a unitary or orthogonal reflection, and it acts on the orthogonal subspace of its reflecting hyperplane H_r^⊥ as some root of unity; we will denote by 𝕌_m the group of m-th roots of unity. As a consequence, the reflections of W sharing the same reflecting hyperplane H (together with the identity) form a cyclic group isomorphic to some 𝕌_m according to their action on H^⊥. One can choose a distinguished generator of this cyclic group, which is the reflection that acts on H^⊥ as multiplication by the root of unity e^2πi/m, and is hence completely determined by H and its order.  This is called the distinguished reflection of H.

From here on, W⊆(V_W) will be a complex reflection group, and V_W will be assumed to be equiped with a W-invariant positive definite hermitian form. We will denote by ℛ the set of reflections of W and by ℋ the set of reflecting hyperplanes of W. To every reflection in ℛ corresponds a hyperplane in ℋ. This correspondence is of course surjective but not injective in general, since all powers of a reflection share the same reflecting hyperplane. 

The group W acts naturally on ℋ, mapping each hyperplane H⊆ V_W to w(H), which will be written as wH. This action corresponds to the action of W on ℛ given by conjugation. Whenever the action on a set X is clear we will denote conjugacy of a,b ∈ X  by a∼ b. Also, ℛ_a→ b will denote the set of reflections mapping a to b.






















 §.§ Transverse hyperplanes


A key notion for the definition of the Brauer-Chen algebra is the notion of transverse hyperplanes. We define them here and gather certain lemmas we will later be using about them.



Let H_1,H_2 be distinct reflecting hyperplanes of W. Then H_1,H_2 are called transverse if they are the only reflecting hyperplanes of W containing their intersection. If this is the case, we write H_1⋔ H_2.  A collection of pairwise transverse hyperplanes of W will be called a transverse collection.



When we say that two reflecting hyperplanes are non-transverse (as opposed to not transverse) it will imply that they are distinct and not transverse; we will denote non-transversality of H_1,H_2 by H_1 ⋔̸H_2. If B is a collection of reflecting hyperplanes of W, we say that H is transverse with B if it is transverse with every element of B, and we write H⋔ B. Correspondingly, we write H⋔̸B if H is non-transverse with some element of B. Finally, if B' is another collection of reflecting hyperplanes, we say that B is transverse with B', and write B⋔ B', if every hyperplane of B is transverse with every hyperplane of B'.

Throughout this work, an important role is played by transverse collections. The action of W on ℋ induces an action on the set of all such collections which will be denoted 𝒞.  When we talk about the orbit or a conjugate of such a collection it will be with respect to this action, and if B is a collection of transverse hyperplanes, then (B) will denote the stabilizer of B under this action. For reasons of uniformity of exposition, we will suppose that 𝒞 contains the empty collection, as well. Furthermore, the empty collection will be assumed to be transverse with every H∈ℋ, and its stabilizer will be W. We denote the set of non-empty transverse collections by 𝒞^*.







If r is a unitary reflection of V_W, then a root of r (or its reflecting hyperplane) is any non-zero vector of the orthogonal complement of its reflecting hyperplane. The following lemma provides a convenient way to check transversality of two hyperplanes using roots. 

  Let H_1,H_2 be reflecting hyperplanes of W with corresponding roots a_1,a_2∈ V_W. Then, H_1,H_2 are transverse if and only if no reflecting hyperplane of W other than H_1,H_2 has a root in the linear span of a_1,a_2.

 If H is a reflecting hyperplane of W with root a then H_1∩ H_2 ⊆ H is equivalent to H^⊥⊆ H_1^⊥⊕ H_2^⊥ which, is in turn, equivalent to a∈⟨ a_1,a_2 ⟩. The result follows.




Let r_1,r_2 be unitary reflections of V_W with respective reflecting hyperplanes H_1, H_2. Then the following are equivalent:


  * r_1,r_2 commute,

  * r_1H_2=H_2,

  * H_1=H_2 or H_1^⊥⊥ H_2^⊥.



For the implication (1) ⇒ (2), if r_1,r_2 commute, then r_1r_2r_1^-1=r_2. Since, r_1r_2r_1^-1 is a reflection with reflecting hyperplane r_1H_2, this means that r_1H_2=H_2.

For (2) ⇒ (3), suppose that r_1H_2=H_2; then r_1H_2^⊥=H_2^⊥. Using the decomposition H_1⊕ H_1^⊥ of V_W and the action of r_1 on H_1, H_1^⊥, one can see that a line of V_W is stable under r_1 if and only if it is contained in H_1 or H_1^⊥. The latter implies that H_2^⊥⊆ H_1, which is equivalent to H_1^⊥⊥ H_2^⊥, and the former that H_1^⊥=H_2^⊥, which is equivalent to H_1=H_2. 

Finally, for (3) ⇒ (1), both of the conditions H_1=H_2 or H_1^⊥⊥ H_2^⊥ imply that r_1,r_2 commute. To see that for the latter, in particular, one may use the decomposition H_1∩ H_2 ⊕ H_1 ^⊥⊕ H_2^⊥ of V_W and the actions of r_1,r_2 on H_1∩ H_2, H_1^⊥, H_2^⊥.



 Let H_1,H_2 be transverse reflecting hyperplanes of W with respective reflections r_1,r_2. Then r_1,r_2 commute.

 If r_1,r_2 do not commute, then, by the previous lemma, r_1H_2 is different from H_2, as well as H_1, since r_1H_2=H_1 would imply that H_2=r_1^-1H_1=H_1 which contradicts transversality of H_1,H_2. But, since r_1 is the identity on H_1∩ H_2 which is contained in H_2, then r_1H_2 contains H_1 ∩ H_2 as well, which is impossible for H_1,H_2 transverse.  









The next two lemmas will be used repeatedly. The first one is  <cit.> and <cit.> and is stated without proof.



Let H_1,H_2 be two distinct hyperplanes in some complex vector space, and s a reflection with reflecting hyperplane H. If sH_1=H_2, then H_1∩ H_2 ⊆ H. Consequently, if H_1,H_2 are transverse reflecting hyperplanes of a complex reflection group W, then there are no reflections in W mapping H_1 to H_2. 




 Let V be a complex vector space, on which we fix a positive definite hermitian form. If H_1^⊥, H_2^⊥⊥ H^⊥, then a unitary reflection s that maps H_1 to H_2 leaves H invariant. This is especially the case when H_1,H_2, H are reflecting hyperplanes of W and H_1,H_2 are transverse with H.

 Let H' be the reflecting hyperplane of s. Since sH_1=H_2, then, by Lemma <ref> above, we have that H_1∩ H_2 ⊆ H' which is equivalent to (H')^⊥⊆ H_1^⊥+H_2^⊥. But, since H_1^⊥,H_2^⊥ are perpendicular to H^⊥, then so is (H')^⊥. By Lemma <ref>, we have that sH=H.


The following lemma will be used only in the very end of this work but we include it here as an application of the two lemmas above.

 Let B={H_1,H_2} and B'={H_1',H_2'} be two disjoint collections of transverse hyperplanes of W. Then, there is at most one reflection mapping H_1 to H_1' and H_2 to H_2'.

 Let s be a reflection mapping B to B' and suppose that sH_1=H_1', sH_2=H_2'; we will show that H_s is determined by H_1,H_2,H_1',H_2'. Hence, if s'H_1=sH_1 and s'H_2=sH_2 for some reflection s' then H_s'=H_s. This implies that both s and s' are powers of some reflection with hyperplane H_s and hence s^-1s' is either such a reflection as well or the identity. Since s'H_2=sH_2, then s^-1s'H_2=H_2 and so, if s^-1s'≠ 1, then we have H_s^⊥⊥ H_2^⊥ or H_s=H_2 (Lemma <ref>) implying, in any case, that sH_2=H_2 which is a contradiction; so s'=s. 


To show now that H_s is uniquely determined, since sH_1=H_1' ≠ H_1, then, by Lemma <ref>, we have H_1∩ H_1' ⊆ H_s and, similarly, H_2∩ H_2' ⊆ H_s. We will show that H_1∩ H_1' ≠ H_2∩ H_2' and hence, since these are subspaces of codimension 1 in H_s, we get that H_s is generated by them, which yields the result. So suppose that H_1∩ H_1' = H_2∩ H_2' =A. Then, A is contained in H_1 ∩ H_2 and it is of codimension 2 in  V, so in fact A=H_1∩ H_2. This yields a contradiction since A is also contained, for example, in H_1' ≠ H_1, H_2, and H_1,H_2 are transverse.
 
 











 §.§ Classification of complex reflection groups


The following classical result is Theorem 1.27 of the book <cit.> of Lehrer and Taylor.

 Every complex reflection group is the product of irreducible complex reflection groups.
















This result reduces the study of complex reflection groups to the study of irreducible ones.
The latter were classified in 1954 by Shephard and Todd <cit.>; they fall into two categories. We give some very introductory information, which is necessary for our analysis later. As usual, the reader is also referred to <cit.>.



  §.§.§ The infinite series



The infinite series is a 3-parameter family of irreducible complex reflection groups denoted by G(m,p,n), where  m,p,n are positive integers, and p|m. The group G(m,p,n) is the subgroup of (ℂ^n) consisting of monomial matrices, i.e. matrices with a unique non-zero entry in every row and every column, whose non-zero entries lie in 𝕌_m, and whose product of entries
lies in 𝕌_m/p. The case where m=p=1 corresponds to the symmetric group S_n acting on ℂ^n by permuting the standard basis. It is the only case when its rank is not n but n-1 (the sum of the vectors of the standard basis is fixed under this action).





  
Reflections and hyperplanes of G(m,p,n).

Let z_1,… z_n denote the standard coordinates of ℂ^n and ζ=e^2πi/m.
The group G(m,p,n) contains at most two classes of distinguished reflections:



  * for 1≤ i≠ j≤ n, and κ∈ℤ,  the reflections (ij)_κ with 

    (ij)_κ (z_1,…, z_i, …, z_j, … ,z_n)= (z_1,…,ζ^κz_j ,… ,ζ^-κz_i,…, z_n)

The reflection (ij)_κ has order 2 and its reflecting hyperplane is  
    H^κ_ij={(z_1,… ,z_n)∈ℂ^n| z_i=ζ^κz_j}.
 When there is no ambiguity as to the system of coordinates, we will say that H^κ_ij has equation z_i=ζ^κz_j.
Note that, (ij)_κ=(ji)_-κ, and, if κ≡ κ'   m, then (ij)_κ=(ij)_κ'. If κ=0 we will just omit it, and write H_ij and (ij) for the corresponding hyperplane and reflection, respectively.
 
 
 
 
 

  * if p ≠ m, then for 1≤ i ≤ n, the reflections t_i with 

    t_i(z_1,… ,z_n)=(z_1,… , ζ^pz_i, … ,z_n).

Each t_i has order m/p, and its corresponding reflecting hyperplane, which we denote by H_i, has equation z_i=0.





  
Transversality relations for G(m,p,n)

The following lemma, the verification of which is left to the reader, sums up all the necessary relations of transversality for the groups of the infinite series.



In G(m,p,n),


  * if i≠ j, then H_i and H_j are non-transverse and the reflections that map one to the other (in any order) are the reflections (ij)_κ, κ∈ℤ;


  * H_i' and H_ij^κ are transverse if and only if i,j  ≠ i'; even in the case where i'∈{i,j}, there exist no reflections mapping one to the other;


  * if j_1≠ j_2, then H_ij_1^κ_1 and H_ij_2^κ_2 are non-transverse and the only reflection mapping one to the other (in any order) is (j_1j_2)_κ_2-κ_1;


  * if {i_1,j_1}∩{i_2,j_2}=∅, then H_i_1j_1^κ_1 and H_i_2j_2^κ_2 are transverse;



  * for (m,p)≠ (2,2), if  i≠ j, then H_ij^κ_1 and H_ij^κ_2 are not transverse and the reflections mapping the former to the latter are  (ij)_κ, for 2κ≡ κ_1+κ_2  m, and t_i^κ_1-κ_2, t_j^κ_2-κ_1, provided that κ_2-κ_1 ≡ 0  p. 


  * if (m,p)=(2,2), then H_ij and H_ij^1 are transverse.  









  §.§.§ The exceptional groups
 

The exceptional complex reflection groups form a family of 34 exceptional groups denoted G_4, … ,G_37 of increasing ranks 2 to 8. They contain the Coxeter groups H_3≅ G_23, F_4 ≅ G_28, H_4 ≅ G_30, E_6 ≅ G_35, E_7 ≅ G_36, E_8 ≅ G_37. Here, we present two examples of non-real exceptional complex reflection groups, to which we will return at the the very end of this article. Again, for a comprehensive exposition of all exceptional groups, the reader is referred to <cit.>, where these groups correspond to the primitive unitary reflection groups.



[Groups G_26,G_25]

The following description of G_26, G_25 can be found, in the form of line systems, in <cit.>, where they correspond to the groups generated by the line systems ℳ_3 and ℒ_3, respectively (see also the table of page 275 of the same book).


Let z_1,z_2,z_3 denote the standard coordinates of ℂ^3 and ζ=e^2πi/3. Assume that ℂ^3 is equiped with the standard inner product. The group G_26 is the subgroup of (ℂ^3), generated by the following 3 types of distinguished unitary reflections:



  * t_i, i=1,2,3, with reflecting hyperplanes H_i with equation z_i=0, and order 3,

  * t_κ,λ, κ,λ = 0,1,2,  with reflecting hyperplanes T_κ,λ with equation z_1 + ζ^κ z_2 + ζ^λ z_3=0, and order 3, and

  * (ij)_κ, for 1≤ i≠ j ≤ 3 and κ=0,1,2, with reflecting hyperplanes H_i,j^κ: z_i=ζ^κz_j, and order 2. If κ=0, we may omit it from the notation.



The group G_25 is the subgroup of G_26 generated by the reflections of the first two types. One may notice that reflections of type (1) and (3) generate the group G(3,1,3) of the infinite series.






























































§ THE BRAUER-CHEN ALGEBRA






 §.§ The definition of the Brauer-Chen algebra


Recall that W is a complex reflection group and ℛ, ℋ denote its sets of reflections and reflecting hyperplanes, respectively.

 Let δ and μ_s, s∈ℛ, be indeterminates, and set μ:={μ_s| s∈ℛ}. The ring of definition of the Brauer-Chen algebra of W is the ring 𝒜=ℤ[μ^± 1,δ^± 1]/(μ_s-μ_s'|  s∼ s'), where ∼ denotes conjugation with respect to W.






The Brauer-Chen algebra associated to W is the 𝒜-algebra (W)
generated by the elements of W subject to the group relations, together with elements e_H, H∈ℋ, subject to the following relations, where H,H_1,H_2∈ℋ and w∈ W:




  (B1) e_H^2=δe_H, 

  (B2) we_Hw^-1=e_wH, 

  (B3) we_H=e_H, if H⊆(w-1),

  (B4) e_H_1e_H_2=e_H_2e_H_1, if H_1⋔ H_2

  (B5) e_H_1e_H_2=∑_s∈ℛ_H_2→ H_1 μ_sse_H_2, if H_1⋔̸H_2

If R is an 𝒜-algebra, we denote the R-algebra R⊗_𝒜(W) by ^R(W). When there is no ambiguity as to R, we will denote the images of δ and μ_s,s∈ℛ in R with the same letters.



With the above convention in mind, ^R(W) is isomorphic to the R-algebra defined by the same generators and relations as in the definition.


 Notice that the condition H⊆(w-1) of relation (B3) is satisfied either when w=1 or when w is a reflection with reflecting hyperplane H. 


 Applying (B2) to the right hand side of relation (B5) of the above definition we get (B5'): e_H_1e_H_2=e_H_1∑_s∈ℛ_H_2→ H_1 μ_ss, if H_1⋔̸H_2.


 It is clear from the definition, that (W) is a quotient of the 𝒜-algebra 𝒜⟨ℋ⟩⋉ W, where 𝒜⟨ℋ⟩ denotes the free associative 𝒜-algebra on ℋ.



  Let R be an 𝒜-algebra. We say that
R is proper (for W) if it has all of the following properties.


  (P1) R is a domain of characteristic zero,

  (P2) (R) has a subfield k_R containing (the image in R of) μ, over which every subgroup of W splits,

  (P3) δ is transcendental over k_R, and (R) is a finite extension of k_R(δ). 



 For any field L of characteristic zero over which every subgroup of W splits (e.g., ℚ or ℂ), if R_0 is a quotient of L[μ^± 1]/(μ_s-μ_s'|  s∼ s') which is a domain, then the ring R=R_0[δ^± 1] is a proper ring of definition, where, for example, one can choose k_R to be the subfield (R_0) of (R). Two notable choices for R_0 are L[μ^± 1]/(μ_s-μ_s'|  s∼ s') and its quotient L[μ^± 1]/(μ_s-μ_s'|  s, s' ∈ℛ).


 Notice that if R is a proper ring of definition, then so is (R). Also, notice that any finite extension of a proper field is also a proper field.


































§ THE SIMPLE MODULES OF ^K(W)







 §.§ Structure of simple modules



Let W be a complex reflection group and K a proper field for W.
In this subsection we obtain some structural properties that simple modules of the Brauer-Chen algebra of W satisfy. We show that every simple module of ^K(W) restricted to KW is isomorphic to a module induced by a simple module of the stabilizer in W of a transverse collection. This spells out the construction of simple modules in the next subsection.

All mentioned modules are finite dimensional. If it is not clear why a constructed module is finite dimensional, we will give an explicit argument. 


For a collection of transverse hyperplanes B we will denote by e_B the element ∏_H∈ B e_H of ^K(W); this product is well defined since e_H,e_H' commute for transverse H, H'. If B is empty, then we assume that e_B=1.


By relation (B2) of ^K(W), it is clear that for w∈ W we have we_Bw^-1=e_wB.


Since we can rearrange the factors in e_B, we have the following multiplication property (see also Definition <ref>). If H∈ B, then e_He_B=δe_B (rearrange the factors of e_B so that e_H is first). If H⋔ B, then B∪{H} is a collection of transverse hyperplanes and e_He_B=e_B∪{H}. Finally, if H⋔̸H' for some H'∈ B, then e_He_B=∑_s∈ℛ_H'→ H μ_sse_B (rearrange the factors of e_B so that e_H' is first). With a symmetric argument we can obtain a similar result for the product e_Be_H, with the only difference arising in the last case, where we get e_Be_H=e_B∑_s∈ℛ_H→ H'μ_ss (see Remark <ref>). 
 
 
 
 
 
 


 Let V be a ^K(W)-module and B a transverse collection. The collection B will be called V-maximal if it is maximal with the property that e_BV≠ 0. The subspace e_BV will be denoted by V_B.
 


 By Remark <ref>, one can see that for w∈ W we have V_wB=wV_B and since the elements w are automorphisms of V, any conjugate of a V-maximal collection is again V-maximal.
 

 Notice that if the empty collection is V-maximal, this means that e_HV=0 for all H∈ℋ.



 Let V be a ^K(W)-module and B a V-maximal collection. Then, for all v∈ V_B and H∈ℋ we have:


    e_Hv=
    δv ,     if H∈ B,
    
    0,     if H ⋔ B, 
    ∑_s∈ℛ_H'→ H μ_ssv,     if H ⋔̸H', H'∈ B




 Since V_B=e_BV, the verification of the formula is immediate from Remark <ref>.






We will show that if V is simple, then it is the direct sum of the subspaces V_B for B in some orbit of V-maximal collections. Before that, we establish existence of a degree on a KW-module V, which will be very useful for the rest of this subsection. Recall from Definition <ref> that K contains a field k_K containing the parameters μ, and over which all subgroups of W split. Also, K is a finite extension of k_K(δ).




Let V be a k_K(δ)W-module. Then, there is a degree map : V→ℤ∪{-∞} such that 
δ^lv =  v + l and xv ≤ v for all v∈ V and x∈ k_KW. Also, v =-∞ if and only if v=0.

 We define the degree of a rational function in δ to be the degree of its numerator minus the degree of its denominator; this does not change if we choose different expressions of the same rational function. It is also clear that this degree satisfies the properties described in the statement.


 Let V be a k_K(δ)W-module. Since W splits over k_K, there exists a k_KW-form of V, namely a k_KW-submodule V' of V such that the natural homomorphism
k_K(δ)⊗_k_KV'→ V is an isomorphism (in fact, this is true for any field k_K, for a proof see <cit.>). Let v_1,… ,v_n be a basis of V'. Then, the aforementioned degree map on k_K(δ) can be extended to k_K(δ) ⊗_k_K V' setting (∑_i^n λ_i ⊗ v_i)=max{ λ_i, i=1,… ,n } for all λ_1,… ,λ_n ∈ k_K(δ). This degree satisfies the properties of the statement. 

To verify  that xv ≤ v for all v∈ V, x∈ k_KW, notice that multiplication on V by elements of k_K does not raise the degree and that the action of a w∈ W on V with respect to the basis 1⊗ v_1, … , 1⊗ v_n is given by a matrix with coefficients in k_K since V' is a k_KW-module.




 Let V be a ^K(W)-module and ℬ an orbit under the action of W on V-maximal collections. The sum of the subspaces V_B, B∈ℬ will be denoted V_ℬ.






Let V be a ^K(W)-module and ℬ an orbit of V-maximal collections. Then,


  * V_ℬ=⊕_B∈ℬV_B,


  * for any B∈ℬ, the set (V_B):={w∈ W|  wV_B=V_B} coincides with (B) (recall that (B) denotes the stabilizer of B in W);


  * V_ℬ is a ^K(W)-submodule of V, and _KW(V_ℬ)≅_K(B)^KW(V_B).








On V, which is naturally a finite dimensional k_K(δ)W-module (recall that K is a finite extension of k_K(δ)), we fix a degree as in Lemma <ref>. Let v=∑_B∈ℬ v_B, with v_B∈ V_B and suppose that v=0. Fix a hyperplane H. The element e_H acts as multiplication by δ on v_B if H∈ B, and by some element of k_KW otherwise, according to Lemma <ref>.

Since v=0, we have that ∑_B∈ℬ, H∈ Bv_B=-∑_B∈ℬ,H∉B v_B. In particular, the degrees of the two sides are equal. Suppose that both sides are non zero. On the left hand side, e_H acts by δ and hence raises the degree by 1, while on the right hand side, acting by some element of k_KW on every v_B with H∉B, it does not raise its degree which leads to a contradiction. So the two sides should be equal to 0 meaning that v=∑_B∈ℬ,H∈ Bv_B=0. Doing the same, now for all H in some fixed collection B∈ℬ, we get that v_B is zero. So, v_B=0 for all B∈ℬ.

Property (1) implies especially that the subspaces V_B are distinct, which implies (2). 


From (1) and (2), it is clear that the KW-module V_B∈ℬ=⊕_ℬ V_B (Definition <ref>) is isomorphic to the induced KW-module _K(B)^KW(V_B), for any B∈ℬ. Also, by Lemma <ref>, every element of ^K(W) acts on each V_B as some element of KW, which implies that V_ℬ is, in fact, a ^K(W)-submodule of V.
 


 Note that the ^K(W)-submodule V_ℬ is completely determined by the K(B)-module V_B: as a KW-module it is induced by V_B, and Lemma <ref> gives the action of the elements e_H,H∈ℋ. We construct all such modules in the next subsection. 












 Let V be a ^K(W)-module and ℬ an orbit of V-maximal collections, such that V=V_ℬ. Then, for every transverse collection C, we have e_CV=⊕_B∈ℬ, C⊆ B V_B. In particular, e_CV≠ 0 if and only if C⊆ B for some B∈ℬ.


From Equation <ref> of Lemma <ref> one can see that for any H and B'∈ℬ we have e_HV_B'⊆⊕_B∈ℬ, H∈ B V_B. We also see that e_H acts as multiplication by δ on ⊕_B∈ℬ,H∈ B V_B. So, since V=⊕_B'∈ℬ V_B', then e_H V=⊕_B∈ℬ,H∈ B V_B. Now, if H_1, H_2 are transverse, then e_H_1e_H_2V=e_H_1V∩ e_H_2V. To see why this is, recall that e_H_1, e_H_2 commute since H_1,H_2 are transverse (Definition <ref>). This implies that e_H_1e_H_2V⊆ e_H_1V∩ e_H_2V. On the other hand, since both e_H_1, e_H_2 act on e_H_1V∩ e_H_2V as scalar multiplcation by δ, we obtain that e_H_1V∩ e_H_2V⊆ e_H_1e_H_2V, which yields equality of the two subspaces. Hence, 


    e_CV=∩_H∈ C (⊕_B∈ℬ,H∈ BV_B)=⊕_B∈ℬ,C⊆ B V_B




 Let V be a ^K(W)-module and ℬ an orbit of V-maximal collections such that V=V_ℬ. Then, ℬ is the only orbit of V-maximal collections. 


By Lemma <ref>, e_CV≠ 0 if and only if C⊆ B for some B∈ℬ. Hence, if C is V-maximal, then C∈ℬ.









Let V be a ^K(W)-module and ℬ an orbit of V-maximal collections.
Then V is simple if and only if V=V_ℬ and V_B is a simple K(B)-module for all B∈ℬ. In this case, it is absolutely irreducible.



 The proof follows the lines of <cit.>. 

By Definition <ref> the subspace V_B and, hence, the ^K(W)-submodule V_ℬ of V are non-zero. So, if V is simple, then V=V_ℬ. Furthermore, if V_B' is a proper and non-zero K(B)-submodule of V_B for some B∈ℬ, then, since every e_H acts on every V_B as an element of KW (Lemma <ref>), the subspace ⊕_w∈ W/(B) wV_B' is stable under e_H and is, hence, a proper and non-zero ^K(W)-submodule of V. So, if V is a simple ^K(W)-module, then V_B is a simple K(B)-module for all V-maximal B. 


For the converse, let V be a ^K(W)-module such that V=V_ℬ. Again, V is naturally a finite dimensional k_K(δ)W-module; fix, hence, a degree on V as in Lemma <ref>.

Suppose that V_B is a simple K(B)-module for some B and let v∈ V_B\{0}. Then K(B)v=V_B, and so ^K(W)v contains ^K(W)V^K_B=V. We will show that for every non zero v∈ V, there exists B∈ℬ such that e_Bv≠ 0. As a consequence, ^K(W)v contains an element of some V_B, in which case, as we just showed, ^K(W)v=V. 

Let v∈ V\{0} and write  v=∑_B∈ℬ v_B where v_B∈ V_B (recall that V=V_ℬ). Let v_B_0 be a term of highest degree. If e_B_0v=0, then:


    e_B_0v_B_0=-∑_B∈ℬ\{B_0} e_B_0 v_B.


Now, since v_B_0 is nonzero, the left hand side is equal to δ^| B_0 |v_B_0 and has a degree of | B_0 | +  v_B_0≠ -∞. Since, for B≠ B_0, e_B_0v_B  has a degree of at most v_B + | B_0 | -1, we see that the degree of the right hand side is strictly smaller than the one on the left, leading to a contradiction. So, e_B_0v≠ 0 and, hence, (W)v=V.

For the absolute irreducibility, suppose that V is simple; hence, V=V_ℬ and V_B is a simple K(B)-module for all B∈ℬ.
We need to show that, for any algebraic extension K' of K, the ^K'(W)-module V':=K' ⊗_K V is simple; it suffices to consider finite extensions. Recall, first of all, that every finite extension of a proper field (Definition <ref>) is again a proper field, so the part of the statement we just proved holds for K' as well, i.e. if ℬ is an orbit of V'-maximal collections such that V'=V'_ℬ and V'_B is a simple K'(B)-module for some B∈ℬ, then V' is simple.

First of all, for all e_H we have e_HV'=e_H(K'⊗_KV)=K'⊗_Ke_HV and, hence, e_BV'=K'⊗_K e_BV. From this it is clear that ℬ is an orbit of V'-maximal collections as well, and that V'_B=K' ⊗_K V_B for all B∈ℬ.
Since V=V_ℬ=⊕_B∈ℬV_B, then V' decomposes as V'=⊕_B∈ℬ (K'⊗_K V_B), and the latter is equal to V'_ℬ by the previous comment. Furthermore, since V_B is a simple K(B)-module and K already contains k_K, over which (B) splits, then V'_B=K'⊗_K V'_B is a simple K'(B)-module. Hence, V' is a simple ^K'(W)-module.
























  
  

 

Let V,V' be two ^K(W)-modules with respective orbits of maximal collections ℬ,ℬ', such that V=V_ℬ and V'=V'_ℬ'. Then V and V' are isomorphic if and only if ℬ=ℬ' and V_B,V_B' are isomorphic K(B)-modules for some (hence, for all) B∈ℬ.

 Let B be a collection of transverse hyperplanes. If the ^K(W)-modules V and V' are isomorphic, then e_BV≠ 0 if and only if  e_BV' ≠ 0, which, by Lemma <ref>, implies that ℬ=ℬ'.
Now if B∈ℬ, it is clear that a ^K(W)-isomorphism V→ V' induces a K(B)-isomorphism V_B=e_BV → e_BV'=V'_B. 

Conversely, as mentioned in Remark <ref> the K(B)-module V_B completetely determines the ^K(W)-module V_ℬ. So, if V_B, V'_B are isomorphic K(B)-modules, for some B, then V,V' are isomorphic ^K(W)-modules. 


The above result implies that simple ^K(W)-modules can be parametrized up to conjugacy by certain pairs (B,V) where B is a collection of transverse hyperplanes and V is a simple K(B)-module. In the next subsection, we describe the pairs that give rise to simple ^K(W)-modules.



































 §.§ Construction of simple modules of ^K(W)




This subsection contains the construction of simple modules, as dictated by the results of the previous subsection, starting with a collection B of transverse hyperplanes and a module V of its stabilizer. The modules V_ℬ are a prototype for this construction. The pairs (B,V) that give rise to such modules are called “admissible”. We also obtain a characterization of these pairs which will be useful for direct calculation.




  §.§.§ Admissibility



For an element y of some KW-module, let _KW(y):={x∈ KW| xy=0} be the annihilator of y inside KW. Recall that 𝒞 denotes the set of collections of transverse hyperplanes.


 Let (B,V) be a pair consisting of a collection B of transverse hyperplanes and a K(B)-module V and let V̂ denote the image of V inside the induced module _K(B)^KWV. The pair (B,V) will be called K-admissible, or just admissible when there is no ambiguity as to the field K, if _KW(e_B)·V̂=0 (here e_B is considered as an element of the KW-module ^K(W)). If there is at least one admissible pair (B,V)  with V≠ 0, then the collection B will be called K-admissible, or just admissible. The set of K-admissible collections will be denoted by 𝒞_adm^K.


 One can see that if B is the empty collection, then all pairs (B,V) where V is a KW-module are K-admissible.







 For H_1,H_2,H∈ℋ, we define the following element of ℤ[μ]W:



    σ^H_H_1,H_2:=∑_s∈ℛ_H_1→ H μ_ss-∑_s∈ℛ_H_2→ H μ_ss.

For B∈𝒞 and H∈ℋ, we define the set

    Σ_B^H:={σ^H_H_1,H_2| H_1,H_2 ∈ B, H_1,H_2⋔̸H}.

Finally, we define the sets 

    Σ_B:=∪_H∈ℋ Σ_B^H   and (B):=(ℛ_B-1)∪ Σ_B,

where ℛ_B:={r∈ℛ| H_r∈ B} and ℛ_B-1={r-1| r∈ℛ_B}.



One can see that wσ^H_H_1,H_2w^-1=σ^wH_wH_1,wH_2 for all
w∈ W and H_1,H_2,H∈ℋ. Furthermore, H_1,H_2 ⋔̸H if and only if wH_1,wH_2 ⋔̸wH. This implies that wΣ_Bw^-1=Σ_wB. It is also  straightforward to verify the same property for the set ℛ_B-1, which yields  w(B)w^-1=(wB).




 Let B∈𝒞. Then, (B)⊆_KW(e_B)

 By defining relation (B3) of ^K(W), we have that re_H_r=e_H_r, for all r∈ℛ. So, if r∈ℛ_B, then rearranging the factors of e_B so that e_H_r comes first, we get that re_B=e_B. This gives the inclusion ℛ_B-1 ⊆_KW(e_B).
Now, recall that if H'∈ B is non-transverse with H, then e_He_B=∑_s∈ℛ_H'→ H μ_sse_B (Remark <ref>). Hence, for H_1,H_2∈ B non-transverse with H, we have σ^H_H_1,H_2e_B=e_He_B-e_He_B=0, i.e. σ^H_H_1,H_2∈_KW(e_B). This gives the inclusion Σ_B⊆_KW(e_B), which concludes the proof.









  §.§.§ The construction of the ^K(W)-module  (B| V)


For the rest of this subsection, we fix B_0∈𝒞 and a K(B_0)-module V_0 with the only assumption that (B_0)V̂_̂0̂=0.

Let ℬ be the orbit of B_0, and set V:=_K(B_0)^KW V_0. Then, V is the direct sum of the subspaces wV̂_̂0̂, w∈ W. Since wV̂_̂0̂ depends only on the left coset w(B_0), or, equivalently, the collection wB_0, we denote it by V^B, where B=wB_0. In particular, V̂_̂0̂=V^B_0. So, we have V=⊕_B∈ℬV^B.

Since  (B_0)V̂_̂0̂=0, by conjugation, we have (B)V^B=0 for all B∈ℬ. Now notice that for every H∈ℋ, the relations Σ_B^H· V^B=0 imply that we can define an operator ε_H,B on V^B by:


    ε_H,Bv:=
    δv,     if  H∈ B,
    
    0,     if  H ⋔ B, 
    ∑_s∈ℛ_H'→ H μ_ss v,     if  H ⋔̸H', H'∈ B.

Since V=⊕_B∈ℬ V^B, this extends to an operator ε_H on V, whose restriction on each V^B equals ε_H,B, for every H∈ℋ. We will show that this makes V a ^K(W)-module. Before that, we have two rather technical but very useful lemmas. For B'∈ℬ let ε_H^(B') be the projection of ε_H onto V^B', i.e. for v∈ V^B,


    ε_H^(B')v:=
    δ_BB'· δv,     if  H∈ B, 
    
    0,     if  H ⋔ B, 
    ∑_s∈ℛ_H'→ H, B→ B' μ_ss v,     if  H ⋔̸H', H'∈ B,

where δ_BB' is the Kronecker delta for B,B', i.e. 0 or 1 if B≠ B' and B=B', respectively.



 Let B,B'∈ℬ be such that ℛ_B→ B'≠∅, and let H,H' be distinct hyperplanes belonging to B,B' respectively. Then, H⋔ H' if and only if H or H' belongs to B∩ B'. 


 Let s∈ℛ_B→ B', and suppose that H'∉B.
If H ⋔ H', then by Lemma <ref>, ℛ_H→ H'=∅, and hence, sH ≠ H'. So, H'=sH” for some H”∈ B other than H. Now, since H”, H' ⋔ H and sH”=H', then, by Lemma <ref>, we have H=sH ∈ B'. 

Conversely, if H∈ B', and H≠ H', then, of course, H⋔ H'. 





 Let B,B'∈ℬ, and H'∈ℋ, such that H'⋔̸B. Then, for all v∈ V^B, we have 

    ε_H'v=∑_s∈ℛ,H'∈ sB1/| B \ sB | μ_ssv,

and 

    ε_Η' ^ (Β')v=δ_H'∈ B'·∑_s∈ℛ_B→ B'1/| B \ B' |μ_ssv,

where δ_H'∈ B' is 1 if H'∈ B' and 0 otherwise.
Finally,  if H_1,H_2∈ B' are non-transverse with B, then
ε_H_1^(B')v=ε_H_2^(B')v.




 

For every H∈ B such that H⋔̸H', of which, by assumption, there is at least 1, we have ε_H'v=∑_s∈ℛ_H→ H' μ_ss v. So, if we denote by N(H',B) the number of H∈ B, H⋔̸H', we get 


    ε_H'v=1/N(H',B)·∑_s∈ℛ, H'∈ sB μ_ssv,

 and Lemma <ref> implies that N(H',B)=| B\ sB|, for any s∈ℛ such that sB contains H'.

So, 
    ε_H'v=∑_s∈ℛ,H'∈ sB1/| B\ sB |μ_ssv

Taking the projection of the two sides onto the subspace V^B' gives:


    ε_H'^(B')v=∑_s∈ℛ_B→ B' ,H'∈ sB1/| B\ sB |μ_ssv=δ_H'∈ B'·∑_s∈ℛ_B→ B'1/| B\ B' |μ_ssv,

with the convention that if B=B' (in which case 1/| B\ B' | is not defined), since Η'∉B, i.e. δ_H'∈ B'=0, the last expression equals 0.


Finally, if H_1,H_2 ∈ B' are not transverse with B, then, applying the last equality for H_1,H_2 gives:


    ε_H_1^ (B')v=∑_s∈ℛ_B→ B'1/| B\ B' |μ_ssv=ε_H_2^ (B')v.







 Mapping e_H↦ ε_H, H∈ℋ makes V a ^K(W)-module, i.e. the operators ε_H satisfy the defining relations of ^K(W):


  (B1) ε_H^2=δε_H,

  (B2) wε_Hw^-1=ε_wH,

  (B3) wε_H=ε_H, if H ⊆(w-1),

  (B4) ε_H_1ε_H_2=ε_H_2ε_Η_1, if H_1⋔ H_2,

  (B5) ε_H_1ε_H_2=∑_s∈ℛ_H_2→ H_1 μ_ss ε_H_2, if H_1⋔̸H_2,

for all w∈ W and H_1,H_2,H ∈ℋ. The ^K(W)-module V is denoted (B_0| V_0) (we remind that the pair (B_0,V_0) is such that (B_0)V̂_0=0).






Notice, first of all, that the definition of ε_H implies that 
ε_HV ⊆⊕_B∈ℬ, H ∈ B V^B. In fact, since 
ε_H acts as multiplication by δ on V^B if H∈ B, we have ε_HV = ⊕_B∈ℬ, H ∈ B V^B, and we also get the first relation. 

For (B2), we verify that it holds on each V^B; this suffices, since V=⊕_B∈ℬV^B. Let B∈ℬ,v∈ V^B and w∈ W. There are three cases to consider. For the first, let wH∈ B, which implies that ε_wHv=δv. Then H∈ w^-1B, and since w^-1v∈ V^w^-1B, we have wε_Hw^-1v=wε_H(w^-1v)=wδw^-1v=δv=ε_wHv. For the second, let wH⋔ B implying that ε_wHv=0. Then, H⋔ w^-1B, and, since w^-1v∈ V^w^-1B, we have wε_Hw^-1v=wε_H(w^-1v)=0=ε_wHv. Finally, for the third case, let wH be non-transverse with some H' ∈ B. Hence, 
ε_wHv=∑_s∈ℛ_H'→ wHμ_ssv. Now, H is non-transverse with w^-1H'∈ w^-1B and, computing as before, we have wε_Hw^-1v=∑_s∈ℛ_w^-1H'→ Hμ_sw^-1swv. As s runs through ℛ_w^-1H→ H the element wsw^-1 runs through ℛ_H'→ wH; also, by their definition μ_s=μ_wsw^-1. So, wε_Hw^-1v=∑_s∈ℛ_H'→ wHμ_ssv=ε_wHv.


For (B3), if H⊆(w-1), and w≠ 1, then w is a reflection with hyperplane H. Now w-1 ∈ℛ_B-1 for every B∈ℬ containing H_w=H, and, since ℛ_B-1 is contained in (B), which, by assumption, annihilates V^B, then (w-1)V^B=0. Since 
ε_HV=⊕_B∈ℬ, H∈ B V^B, this gives (w-1)ε_H=0, or wε_H=ε_H.

For (B5), if H_1⋔̸H_2, then, for every B∈ℬ containing H_2 and v∈V̂^B, we have ε_H_1v=∑_s∈ℛ_H_1→ H_2 μ_ssv. Since ε_H_2V=⊕_B∈ℬ, H_2∈ B V^B, the property follows.


We show now (B4), i.e. ε_H_1ε_H_2=ε_H_2ε_H_1 if H_1⋔ H_2, proving the relation on V^B, for every B∈ℬ. 
For the rest of the proof, fix a B∈ℬ and a v∈ V^B.
We distinguish four cases. The proof of the first three is relatively short.



  * H_1∈ B, which implies that ε_H_1v=δv. If H_2⋔ B or H_2∈ B, then ε_H_2v=0 or δv, respectively; in any case, it commutes with ε_H_1. So suppose that H_2 is non-transverse with some H∈ B, which implies that
ε_H_2v=∑_s∈ℛ_H→ H_2μ_ssv.  Now,  since H, H_2 ⋔ H_1, then, by Lemma <ref>, for all s∈ℛ_H→ H_2, we have H_1=sH_1∈ sB. Since sv∈ V^sB, this implies that ε_H_1(sv)=δsv for all s∈ℛ_H→ H_2, and, hence, ε_H_1ε_H_2v=δ∑_s∈ℛ_H→ H_2μ_ssv=ε_H_2δv=ε_H_2ε_H_1v. 


  * H_1⋔ B, which implies that ε_H_1v=0. As before, if H_2⋔ B or H_2∈ B, then ε_Η_2v=0 or δv, respectively, and ε_H_1,ε_H_2 commute.

So, suppose H_2⋔̸H∈ B, and, hence, ε_H_2v=∑_s∈ℛ_H→ H_2μ_ssv. As before, by Lemma <ref>, since H,H_2⋔ H_1, if s∈ℛ_H→ H_2, then sH_1=H_1, which implies that H_1=sH_1 ⋔ sB. Hence, ε_H_1V^sB=0, for all s∈ℛ_H→ H_2, and, so, ε_H_1ε_H_2v=0=ε_H_2ε_H_1v.




  * There are H_1',H_2'∈ B, such that H_1' is non-transverse with H_1 but transverse with H_2, and H_2' is non-transverse with H_2 but transverse with H_1. Then, ε_H_1v=∑_s∈ℛ_H_1' → H_1μ_ssv and ε_H_2v=∑_s∈ℛ_H_2'→ H_2μ_ssv.  Again, by Lemma <ref>, since H_1',H_2 ⋔ H_2', if  s∈ℛ_H_1' → H_1, then Η_2'=sH_2' ∈ sB, and, so, ε_H_2sv=∑_s'∈ℛ_H_2'→ H_2μ_s's'sv. Hence,
 
 
    ε_H_2ε_H_1v=∑_s∈ℛ_H_1'→ H_1
    s'∈ℛ_H_2'→ H_2 μ_sμ_s'ss' v.
 With the same reasoning, we obtain the same result for 
 ε_H_1ε_H_2v.






So now, we can assume that there is H∈ B non-transverse with both H_1 and H_2. To lighten the notation, for the rest of the proof, s and s' will always denote reflections, and we will omit mentioning s,s'∈ℛ, which will be assumed for every sum. Also, for H'∈ℋ let ℬ_H' be the set of elements of ℬ that contain H'.
We will be using the results of Lemma <ref> repeatedly. The rest of the proof is rather long but most of it is a detailed  regrouping of certain sums.
We have:





    ε_H_2ε_H_1v       =ε_H_2∑_sB∈ℬ_H_11/| B\ sB | μ_ssv= ∑_sB∈ℬ_H_1∩ℬ_H_2δ/| B\ sB | μ_ssv 
    +∑_sB∈ℬ_H_1\ℬ_H_2
     s'sB∈ℬ_H_1∩ℬ_H_2μ_s'μ_s/| B\ sB || sB\ s'sB |s'sv.
        =∑_sB ∈ℬ_H_1∩ℬ_H_2δ/| B\ sB | μ_ssv + Σ_1 v,

where Σ_1:=∑_sB∈ℬ_H_1\ℬ_H_2
 s'sB∈ℬ_H_1∩ℬ_H_2μ_s'μ_s/| B\ sB || sB\ s'sB |s's. Similarly, we compute,



    ε_H_1ε_H_2v    =∑_sB ∈ℬ_H_1∩ℬ_H_2δ/| B\ sB | μ_ssv + Σ_2v,

where Σ_2:=∑_sB∈ℬ_H_2\ℬ_H_1
 s'sB∈ℬ_H_1∩ℬ_H_2μ_s'μ_s/| B\ sB || sB\ s'sB |s's.


The first term in both sums being the same, we need to show that Σ_1v=Σ_2v. For convenience, let 
f(s',s):=μ_s'μ_s/| B\ sB || sB\ s'sB |s's,
which makes sense as long as B≠ sB and sB≠ s'sB.

The following rearrangement of Σ_1 just amounts to writing ℬ_H_1\ℬ_H_2 =ℬ_H_1\ (ℬ_H_1∩ℬ_H_1) while adding the condition s'sB≠ sB, which is true for the set of summation of Σ_1, but not for the following sums. We have:


    Σ_1=∑_sB∈ℬ_H_1\ℬ_H_2
     s'sB∈ℬ_H_1∩ℬ_H_2
     f(s',s)   = ∑_sB∈ℬ_H_1
      s'sB ∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sB f(s',s) - ∑_sB ∈ℬ_H_1∩ℬ_H_2
    s'sB ∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sB f(s',s) =S_1-∑_sB∈ℬ_H_1∩ℬ_H_2
     s'sB∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sB f(s',s),

where S_1:=∑_sB∈ℬ_H_1
  s'sB ∈ℬ_H_1∩ℬ_H_2
s'sB≠ sB f(s',s),
and, in the same way Σ_2 = S_2 - ∑_sB ∈ℬ_H_1∩ℬ_H_2
 s'sB ∈ℬ_H_1∩ℬ_H_2
s'sB≠ sB f(s',s),
where S_2:=∑_sB∈ℬ_H_2
 s'sB∈ℬ_H_1∩ℬ_H_2
s'sB≠ sB  f(s',s).



Again, the second terms of the final expressions for Σ_1,Σ_2 are the same, and, so, it suffices to show that S_1v=S_2v. We have:


    S_1v=∑_sB∈ℬ_H_1
     s'sB ∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sBμ_s'μ_s/| B\ sB || sB\ s'sB |s's v =∑_sB∈ℬ_H_1
     s'sB ∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sBμ_s's'/| sB\ s'sB|·μ_ss/| B\ sB | v


Now, we regroup the reflections s in the last sum with respect to the set sB, introducing the variable b=sB. We have:



    S_1v   =∑_b∈ℬ_H_1( ∑_s'b ∈ℬ_H_1∩ℬ_H_2
     s'b≠ bμ_s's'/| b\ s'b|·∑_sB=bμ_ss/| B\ b |v )=∑_b∈ℬ_H_1( ∑_s'b ∈ℬ_H_1∩ℬ_H_2
    s'b≠ bμ_s's'/| b \ s'b |·∑_sH=H_1, sB=b μ_ssv)  
        =∑_b∈ℬ_H_1∑_sH=H_1 
     sB=b 
     s'sB∈ℬ_H_1∩ℬ_H_2
    s'b≠ bμ_s's'/| b \ s'b |μ_ssv=∑_sH=H_1
     s'sB∈ℬ_H_1∩ℬ_H_2
    s'sB≠ sBμ_s's'/| sB \ s'sB | μ_ss v



Again, we just perform two regroupings of the terms in the last sum with respect to the sets s'sB=b' and s^-1b'=b”:


    S_1v    =∑_b'∈ℬ_H_1∩ℬ_H_2∑_sH=H_1
     sΒ ≠ b'  
     s'sB=b'μ_s's'/| sB\ b' | μ_ss v=∑_b'∈ℬ_H_1∩ℬ_H_2∑_b”∈ℬ∑_sH=H_1 
    sB ≠ b' 
     sb”=b'
     s'sB=b'μ_s's'/| sB\ b' | μ_ssv
       =∑_b'∈ℬ_H_1∩ℬ_H_2∑_b”∈ℬ
    b”≠ B∑_sH=H_1 
     sb”=b'
     s'sB=b'μ_s's'/| sB\ b' | μ_ssv.



For the passage to the last sum, since in the innermost summation, the third condition is sb”=b', then the second condition, i.e. sB≠ b' becomes b'≠ B, which we can take out to the second sum as we did in the last expression. In the same way for S_2, we get:


    S_2v=∑_b'∈ℬ_H_1∩ℬ_H_2∑_b”∈ℬ
     b”≠ B∑_sH=H_2 
     sb”=b'
     s'sB=b'μ_s's'/| sB\ b' | μ_ssv.


Now, since in both expressions the first and second summations are the same, it is enough to show that the innermost sums are equal; i.e. that for all b'∈ℬ_H_1∩ℬ_H_2, and b”∈ℬ with b”≠ B, we have



    ∑_sH=H_1
     sb”=b' 
     s'sB=b'μ_s's'/| sB \ b' | μ_ss v=∑_sH=H_2
     sb”=b' 
     s'sB=b'μ_s's'/| sB \ b' | μ_ss v.



Notice first that, on both sides, if the summation is not empty, then H∈ b”.
Now, we take the left-hand side, and see that


    ∑_sH=H_1
     sb”=b' 
     s'sB=b'1/| sB \ b' | μ_s's'μ_ss v     =∑_sH=H_1
     sb”=b' 
     s'sB=b'1/| sB \ b' | μ_s'μ_ss(s^-1s's) v =∑_sH=H_1
     sb”=b' 
     aB=b”1/| B \ b”| μ_aμ_s sav 
       =∑_sH=H_1
    sb”=b'μ_ss∑_aB=b”μ_aa/| B\ b”|  v=∑_sH=H_1
    sb”=b'μ_ssv_0

where v_0=∑_aB=b”μ_aa/| B \ b”|v.
For the passage to the second line, when s' runs through all reflections that take sB to b', then a=s^-1s's runs through all reflections mapping B to b”; also, μ_a=μ_s', since a and s' are conjugate, and | sB\ b' | = | s^-1sB \ s^-1b'| =| B\ b”|.

Correspondingly, the right-hand side of Equation (<ref>) equals ∑_sH=H_2
sb”=b'μ_ssv_0.

Now, notice that v_0∈ V^b” and recall that H∈ b”. So, by Lemma <ref>, 
∑_sH=H_1
sb”=b'μ_ssv_0=ε_H_1^(b”)v_0 which, by the same lemma equals ε_H_2^(b”)v_0=∑_sH=H_2
sb”=b'μ_ssv_0, since H∈ b” is not transverse with H_1,H_2. This concludes the proof.









 If V_0≠ 0, then the transverse collection B_0 is a (B_0| V_0)-maximal collection, for which (B_0| V_0)_B≅_K(B_0) V_0. Furthermore, (B_0| V_0)=(B_0| V_0)_ℬ (see Definition <ref>).

 The element e_H acts on (B_0| V_0) as ε_H, and, as we remarked in the beginning of the proof of the previous proposition, ε_H(B_0| V_0)=⊕_B∈ℬ, H∈ BV^B. Now since e_H_1,e_H_2 commute for H_1⋔ H_2, then, for any B'∈𝒞, we have e_B'(B_0|V_0)=∩_H∈ B'e_H(B_0| V_0)=⊕_B∈ℬ, B'⊆ B V^B. In particular, for B∈ℬ we have e_B(B_0| V_0)=V^B, and, so e_B_0(B_0| V_0)=V^B_0=V̂_̂0̂≅_K(B_0)V_0. Also, for all B'∈ℬ such that B_0⊂ B' we get
e_B'(B_0| V_0)=0, which implies that B_0 is (B_0| V_0)-maximal. Finally, (B_0| V_0)=⊕_B∈ℬ V^B=⊕_B∈ℬ (B_0|V_0)_B=(B_0|V_0)_ℬ.


 One can see from the above proposition and Remark <ref> that if B_0 is the empty collection, this construction turns all KW-modules into ^K(W)-modules where all e_H, H∈ℋ act as zero.  





 §.§ Conclusions


In this subsection, we combine the results of Subsections <ref> and <ref> and obtain the classification of simple ^K(W)-modules.  

 Let B∈𝒞 and V a K(B)-module. Then there exists a ^K(W)-module V' with maximal collection B and such that V'_B is a K(B)-module isomorphic to V if and only if (B,V) is admissible. Furthermore, (B,V) is admissible if and only if (B)V̂=0.

  If (B,V) is admissible, i.e. _KW(e_B)V̂=0, then, since (B)⊆_KW(e_B), we get that (B)V̂=0. In that case, the ^K(W)-module (B|V) satisfies the properties of the statement, by Proposition <ref>. Conversely, suppose that a module V', as in the first statement, exists. Since V_B=e_BV, then _KW(e_B)V_B=0, and, since V_B⊆ V_ℬ=⊕_w∈ W/(B)wV_B, where ℬ is the W-orbit of B (Proposition <ref>), then _KW(e_B)V̂_B=0. Since V_B and V are isomorphic as K(B)-modules, we have _KW(e_B)V̂=0, and, hence (B,V) is admissible. 

For the second part of the statement, since (B)⊆_KW(e_B), then if (B,V) is admissible we have (B)V̂=0. For the converse, if (B)V̂=0, then the module (B| V) satisfies the properties of the statement, and hence (B,V) is admissible.





	

Proposition <ref> tells us that the module (B| V) satisfies the conditions for most of the results of Subsection <ref>. 



 Let (B,V) be an admissible pair. Then the ^K(W)-module (B| V) is simple (in which case it is, also, absolutely irreducible) if and only if V is a simple K(B)-module.

 By Proposition <ref>, the ^K(W)-module (B| V) satisfies the conditions of Proposition <ref>, and the result follows from the latter proposition.














 Let V be a ^K(W)-module, and ℬ an orbit of V-maximal collections. Then, the pair (B,V_B) is admissible and  the ^K(W)-module V_ℬ is isomorphic to (B| V_B), for every B∈ℬ. As a consequence, V contains a module isomorphic to (B| V_B).

 Proposition <ref> implies that the pair (B,V_B) is admissible. Now, both modules V_ℬ and (B| V_B) satisfy the conditions of Proposition <ref> (the latter by Proposition <ref>), which implies the result.



The next theorem may be considered as the central result of this section. When we say that two pairs (B,V),(B',V') are conjugate, we mean that there is w∈ W such that wB=B' and also wρ_Vw^-1 is isomorphic to ρ_V', where ρ_V,ρ_V' are the  representations corresponding to V,V', respectively.


The simple ^K(W)-modules are the modules (B| V) where (B,V) is an admissible pair and V a simple K(B)-module; they are, in that case, absolutely irreducible. Two such ^K(W)-modules (B| V) and (B'|V') (in fact, not necessarily simple) are isomorphic if and only if (B,V) and (B',V') are conjugate. 

 By Proposition <ref>, a ^K(W)-module V' is simple and, in fact, absolutely irreducible, if and only if V'=
V'_ℬ for an orbit ℬ of V'-maximal collections and V'_B is a simple K(B)-module for some B∈ℬ. Propositions <ref> and  <ref> yield, now, the first part of the theorem. 

For the second statement,  since, by Proposition <ref>, the ^K(W)-module (B| V), for any admissible pair (B,V) satisfies the conditions of Proposition <ref> the result follows from the latter proposition.




 If (B,V) is an admissible pair, then, counting V' up to K(B')-isomorphism, there are exactly | W ||(B) | admissible pairs (B',V') for which (B'| V')≅ (B| V).

 This is implied by the second statement of the above theorem. 



 Let B be a transverse collection and V_i, i∈ I be K(B)-modules. Let V=⊕_i∈ IV_i Then, (B,V) is admissible if and only if (B,V_i) is admissible for all i∈ I, in which case:

    (B| V)≅⊕_i∈ I(B| V_i).

In particular, if V_i is simple for all i∈ I, then the above sum gives the decomposition of (B| V) into simple ^K(W)-modules.


 Verifying that “(B,V) is admissible if and only if each (B,V_i) is”, is straighforward using the definition. 
By Proposition <ref>, B is a (B| V_i)-maximal collection and (B| V_i)_B≅ V_i. Again, one can see that B is also ⊕_i∈ I(B| V_i)-maximal, and 
⊕_i∈ I(B| V_i)_B=e_B(⊕_i∈ I(B| V_i))=⊕_i∈ Ie_B(B| V_i)≅⊕_i∈ IV_i.
Since this is the case for (B| V) as well, the isomorphism (<ref>) is implied by Proposition 
<ref>. 





























§ SEMISIMPLICITY


In this section, we show that the Brauer-Chen algebra is split-semisimple over
any proper field (Definition <ref>). We obtain a lower bound for the dimension of ^K(W), given the simple 
modules we constructed in the previous section, after a necessary analysis on
admissibility, which will be extensively used for all that will follow.
Next we find an adequate upper bound for the dimension, and prove that the algebra is semisimple.
Finally, we give a decomposition of the simple modules of ^K(W) with respect to 
the Brauer-Chen algebra of a parabolic subgroup.





 §.§ Analysis of admissibility


For this subsection, we fix a transverse collection B∈𝒞. Recall that for a K(B)-module V we denote by V̂ its image in the induced KW-module _K(B)^KW(V). Also, W/(B) will denote a set of left coset representatives of (B) in W. 

 For w∈ W, let π_w be the projection KW→ wK(B) with respect to the decomposition KW=⊕_w'∈ W/(B) w'K(B), i.e. if x=∑_h∈ Wλ_hh ∈ KW, then π_w(x):=∑_h∈ w(B)λ_hh.



 Suppose that B'∈ℬ and denote  by π'_w the corresponding projection KW→ wK(B'). One can verify that if w'B=B', then w'π_w(x)(w')^-1=π' _w'w(w')^-1(w'x(w')^-1), for all x∈ KW.


 Let V be a K(B)-module and x∈ KW. Then xV̂=0 is equivalent to π_w(x)V̂=0 for all w∈ W, or to w^-1π_w(x)V=0 for all w∈ W (the last expression makes sense since w^-1π_w(x)∈ K(B)).

 This is immediate from the decomposition _K(B)^KW(V)=⊕_w∈ W/(B) wV̂.



Recall that a pair (B,V) is admissible if _KW(e_B)·V̂=0, or equivalently, by Proposition <ref>, if (B)V̂=0.
Based on the above lemma, we obtain alternative characterisations of admissibility. We group them all together in Proposition <ref>, after establishing some terminology.

 We define the following sets:

    _KW(B):=∑_w∈ Wπ_w(_KW(e_B)),   (B):=∪_w∈ Wπ_w((B)),



    _K(B)(B):=∑_w∈ Ww^-1π_w(_KW(e_B))






In view of Remark <ref>,
the properties w_KW(e_B)w^-1=_KW(e_wB) and w(B)w^-1=(wB) imply that conjugation by W on the above sets commutes with the action of W on 𝒞, i.e., for example, w_KW(B)w^-1=_KW(wB).




 For all w,w'∈ W, s∈(B) and x∈ KW, we have π_w(w'xs)=w'π_(w')^-1w(x)s.

 Let x=∑_h∈ Wλ_hh∈ KW and w,w'∈ W, s ∈(B). We have 

    π_w(w'xs)   =
    π_w(∑_h∈ Wλ_hwhs)=∑_w'hs∈ w(B)λ_hw'hs =∑_h∈(w')^-1w(B)λ_hw'hs=w'π_(w')^-1w(x)s





 The set _KW(B) is a left KW-ideal which is also stable under right multiplicatiοn by (B). The set _K(B)(B) is a two-sided K(B)-ideal. Finally, we have

    _KW(B)=⊕_w∈ W/(B)w_K(B)(B).



 The annihilator _KW(e_B) is a left ideal of KW stable under right multiplication by (B); together with Lemma <ref> above, this implies the first statement.

 For the second statement, by the same lemma, for w∈ W and s, s'∈(B) we have 
    sw^-1π_w(_KW(e_B))s'=π_s(sw^-1_KW(e_B)s').
 Summing over all w∈ W, since sw^-1_KW(e_B)s'=_KW(e_B), we get that _K(B) is stable under left and right multiplication with (B).
 
  Finally, in the same way, for all w' ∈ W we have w'w^-1π_w(_KW(e_B))=π_w'(w'w^-1_KW(e_B)). Again, since w'w^-1_KW(e_B)=_KW(e_B), summing over all w,w'∈ W yields that 
    W·_K(B)(B)=_KW(B).
 Since _K(B)(B) lies inside K(B), this implies that  
    _KW(B)=⊕_w∈ W/(B)w_K(B)(B).







 Let V be a K(B)-module.
The following statements are equivalent:


  (1) (B,V) is admissible, i.e., _KW(e_B)V̂=0,

  (2) _KW(B) V̂=0,

  (3) _K(B)(B) V=0.

  (4) (B) V̂=0,

  (5) (B) V̂=0.



 As mentioned earlier, the equivalence of assertions (1) and (4) is given by Proposition <ref>. Lemma <ref> yields the equivalences (1)⇔ (2) ⇔ (3) and (4) ⇔ (5).




 The following statements are equivalent:


  * B is admissible,

  * _K(B)(B)≠ K(B),

  * _KW(B)≠ KW
 

 The equivalence of (2) and (3) is immediate from the definitions of _K(B)(B) and _KW(B) and (1)⇔ (2) is implied by the equivalence (1)⇔ (3) of Proposition <ref>.









Τhe following proposition provides an explicit relation of _KW(B) and _K(B)(B) with the sets (B), (B). 

 Let I denote the two-sided ideal of K(B) generated by the union of the sets w^-1π_w((B)), w∈ W (for this proposition only). Then,
 

  * _K(B)(B)=I,

  * _K(B)(B)=π_1(KW·(B)),

  * _KW(B)=KW·(B).



 By Proposition <ref>, for any K(B)-module V, the condition (B)V̂=0 is equivalent to π_w((B))·V̂=0, for all w∈ W. The latter is in turn equivalent to w^-1π_w((B))· V=0, for all w∈ W or, by definition of I, to IV=0. Since, by (4)⇔ (3) of the same proposition, this is equivalent to _K(B)(B)V=0 and I,_K(B)(B) are both two-sided ideals of K(B), considering their corresponding quotients as the K(B)-module V, we get (1). 


For (2), by (1) we get that _K(B)(B) is equal to 

    K·∑_s,s'∈(B),w∈ Wsw^-1π_w((B))s'.

By Lemma <ref> we can write this as

    K·∑_s,s'∈(B),w∈ Wπ_s(sw^-1(B)s').

Now, for s∈(B) we have π_s=π_1, and, since (B) is stable under conjugation by (B), we can write sw^-1(B)s'=sw^-1s'(B). In the above sum, the element sw^-1s' runs through all elements of W, which yields (2).

Finally, Proposition <ref> implies that _KW(B)=W _K(B)(B). By (2), we have that _K(B)(B)=∑_w,w'∈ Wwπ_1(Kw'(B)). By Lemma <ref> again, this is 

    ∑_w,w'∈ Www'π_(w')^-1(K(B))
 which one can see that equals KW·(B).












 §.§ Lower bound for the dimension



 Recall that 𝒞 denotes the set of transverse collections and 𝒞^K_adm the subset of admissible ones. Also the notation (K(B)) will be used for the set of simple modules of K(B).


By Theorem <ref>, the simple modules of ^K(W) are the modules (B|V), where (B,V) is an admissible pair, which are, in fact, absolutely irreducible. By Corollary <ref>, there are exactly | W | / |(B) | such pairs for every isomorphism class of ^K(W)-modules. Hence, the sum of squares of dimensions of the simple ^K(W)-modules equals:


    ∑_B∈𝒞,V ∈(K(B))
     (B,V) adm. |(B) |/| W|(_K (B|V))^2.

Recall that the module (B|V) is KW-isomorphic to _K(B)^KW(V). Hence, its dimension equals | W ||(B) |·_K V.
Consequently, for any B∈𝒞 we have,



    ∑_ V ∈(K(B))
     (B,V) adm.  (_K (B|V))^2
       =(| W |/|(B) |)^2 ∑_ V ∈(K(B))
     (B,V) adm.  (_K V)^2



Now, by characterisation (3) of admissibility of Proposition <ref>, a pair (B,V) is admissible if and only if the K(B)-module V factors through the quotient algebra K(B)/_K(B)(B). Since the latter, as a quotient of K(B), is split-semisimple over K (see Definition <ref>), then its dimension equals the sum of squares of dimensions of its simple modules. Combining these two properties, for any B∈𝒞, we have: 


    ∑_V∈(K(B))
    (B,V) adm. (_K V)^2    = _K(K(B)_K(B)(B))=|(B)|/| W |·_K (KW_KW (B)).

This implies that 

    ∑_V∈(K(B))
    (B,V) adm. ( (B|V))^2
       =| W |/|(B)|·_K(KW_KW B).

Replacing this in the sum (<ref>), we obtain the following result.


 The sum of squares of dimensions of all absolutely irreducible ^K(W)-modules equals:

    ∑_B∈𝒞_K(KW_KW B).









 §.§ Upper bound and semisimplicity


We prove now that the Brauer-Chen algebra is semisimple by bounding its dimension from above. The following lemma is a slight improvement of  <cit.> which can be seen to be true with essentially the same proof. In any case, we give a quick proof here as well. 

  The elements we_B, where w∈ W and  B∈𝒞, span (W) as an 𝒜-module.

 Remark <ref>, giving a formula for the product e_He_B, implies that the 𝒜-span of the elements in the statement, is, in fact, a left ideal of (W). Since they contain 1, the result follows. 



 Let R be an 𝒜-algebra. For r ∈ℕ, we define I_r^(R):=∑_B∈𝒞, | B |≥ r RWe_B⊆^R(W). If R=𝒜 we omit it from the notation and write I_r. 


 Let R be an 𝒜-algebra. The set I_r^(R) is a two-sided ideal of ^R(W).

 By definition, RW· I_r^(R)=I_r^(R). Since we_B=e_wBw for any w∈ W and B∈𝒞, then I_r^(R)KW=I_r^(R) and in fact, I_r^(R)=∑_B∈𝒞, | B |≥ r e_BRW. Combining the latter with the formula for the product e_He_B of Remark <ref>, one can see that e_HI_r^(R)⊆ I_r^(R) for all H∈ℋ. In the same way, using the formula for e_Be_H of the same remark, we get that I_r^(R)e_H ⊆ I_r^(R) as well.


The ideals I_r have already been introduced in <cit.> , and in the same paper the quotient algebra (W)/ I_r+1 is denoted _r(W). The following lemma will provide our setting for all situations where we want to suitably restrict the spanning set of Lemma <ref>.

 Let R be an 𝒜-algebra. The ideals I^(R)_r, r∈ℕ form a descending chain of ideals of ^R(W), such that ^R(W)=∪_r∈ℕI^(R)_r.

 It is clear by the definition that I^(R)_r+1⊆ I^(R)_r for all r∈ℕ. Furthermore,  Lemma <ref> is equivalent to the property ^R(W)=∪_r∈ℕI^(R)_r.




For any B∈𝒞, we have _KW (B) e_B ⊆ I_| B |+1^(K).




Let r=| B |.
If e_B ∈ I_r+1^(K), then the result is immediate, since ΚW I_r+1^(K)⊆ I_r+1^(K). So, we suppose that e_B ∉I_r+1^(K). Consider the quotient _r^K(W), and let a∈ KW. Of course, ae_B=0 in _r^K(W) is equivalent to ae_B∈ I_r+1^(K). We will show that in _r^K(W),  ae_B=0 implies  π_w(a)e_B=0 for all w∈ W. Hence, since _KW(B)=∑_w∈ Wπ_w(_KW(e_B)), we get the statement.

Let ℬ denote the orbit of B, and let V=∑_B'∈ℬ e_B'(KW + I_r+1^(K)) ⊆_r^K(W). Relation (B2) of Definition <ref> or Remark <ref> imply that wV=V for w∈ W. Furthermore, if H is a hyperplane and B'∈ℬ, Remark <ref> yields that e_He_B' either belongs to KWe_B'⊆ V, or is equal to some e_B”, where B” strictly contains B', in which case e_B”∈ I_r+1^(K). So, e_HV⊆ V for all hyperplanes H, as well, and hence, V is a ^K(W)-module.

Now, it is clear that (e_B + I_r+1^(K))KW ⊆ e_BV since e_B (e_Bx+I_r+1^(K))=δ^r (e_Bx + I_r+1^(K)) for any x∈ KW. In particular, e_BV≠ 0. Furthermore, B is V-maximal since if B' strictly contains B, then e_B'∈ I_r+1^(K), and so, e_B'V=0. Now Proposition <ref> yields that ∑_B'∈ℬ V_B'=⊕_B'∈ℬ V_B'. Since wV_B=V_wB for all w∈ W, this implies that for every a∈ KW and v∈ V_B, av=0 implies π_w(a)v=0. So, ae_B=0 in ^K_r(W) implies π_w(a)e_B∈ I_r+1^(K) for all a∈ KW and w∈ W and the result follows.










 The algebra ^K(W) is split-semisimple, with dimension


    ∑_B∈𝒞_K (KW_KW(B))  or, equally, ∑_B∈𝒞^K_adm_K (KW_KW(B)).







By Lemma <ref>, we have that ^K(W)=∪_r∈ℕI^(K)_r. We write I^(K)_r as ∑_| B | =r KWe_B + I^(K)_r+1. By Lemma <ref> above, we have _KW(B)e_B⊆ I_r+1^(K) for all B of cardinality r, which implies that 


    _K I^(K)_r ≤∑_| B | =r_K(KW_KW (B)) + _K I^(K)_r+1.

By induction on r, this yields


    _K^K(W)≤∑_B∈𝒞_K(KW_KW (B)).

The above sum is equal to the the sum of squares of dimensions of simple ^K(W)-modules (Lemma <ref>), which yields the main result. Equality with the second sum of the statement is implied by Corollary <ref>, which states that if B is not admissible, then _KW(B)=KW.



 The dimension of ^K(W) is equal to:


    ∑_B∈𝒞^K_adm| W |/|(B) |·_K(K(B)_K(B)(B)).


 The statement is implied by Theorem <ref>, since 
_KW(B)=⊕_w∈ W/(B)w_K(B)(B), and, hence, 

    _K (KW_KW(B))=| W |/|(B) |_K(K(B)_K(B)(B)).




 In the case of the empty collection B, one can verify that _K(B)(B)=0. This, as expected (see also Remark <ref>), implies that the part of (<ref>) in the above corollary corresponding to B equals 
the order of W. 














































§ ADMISSIBILITY AND A BASIS FOR ^K(W)




We apply now the established results of the previous sections to obtain, for all irreducible complex reflection groups, the admissible
collections and corresponding admissible pairs. Following the results of the previous section, for a transverse collection B, these elements are determined by the ideal _K(B)(B) of K(B). 

Case-by-case analysis shows that for every admissible transverse collection B, there is
a certain subgroup K_B of (B) such that _K(B)(B) is
some kind of modified augmentation ideal of this subgroup. In fact, for all but two groups (and specific proper fields of definition), it is equal to the augmentation ideal of K_B. This translates, equivalently, into the fact that the admissible
K(B)-modules V are those that induce a certain representation on K_B, which,
again, for most cases, is the trivial one. 



 §.§ Overview of this section's results


We sum up, in this first subsection, the results that we obtain throughout the rest of this section by the case-by-case study of all irreducible complex reflection groups, and provide some general conclusions as well.
  
For the next definition, recall that for B∈𝒞, ℛ_B={r∈ℛ| H_r∈ B}.

 Let W be a complex reflection group and B a transverse collection. We denote by K_B the subgroup of (B) generated by ℛ_B together with all products s_2^-1s_1 where s_1,s_2∈ℛ satisfy: 


  (c1) s_1B=s_2B≠ B (in particular, s_2^-1s_1∈(B)),

  (c2) s_1H≠ s_2H, for all H∈ B\ s_1B.

If B is empty, we define K_B=1.


 One can verify that for w∈ W and B∈ℬ we have wK_Bw^-1=K_wB. In particular, K_B is a normal subgroup of (B).


 If B is a transverse collection of cardinality 1, i.e., B consists of a single hyperplane H, the above definition of K_B yields the subgroup of W generated by ℛ_B, which coincides with the pointwise stabilizer W_H of H. 


For a function χ: K_B→ K, we denote by ^χ(K_B) the two-sided ideal
of K(B) generated by all h-χ(h), h∈ K_B. Note that if χ=1, this is the 
usual augmentation ideal (K_B). The following proposition is proved throughout the course of this section.


 Let W be an irreducible complex reflection group and K a proper
field. Then, if B∈𝒞^K_adm, there is a group homomorphism χ:K_B→𝕌_6 ∩ K
such that _K(B)(B)=^χ(K_B). Furthermore, if W≠ G_25,G_32, then χ=1. If W=G_25 or G_32, then _K^χ(K_B)=_K (K_B). 


A consequence of this proposition is the following result, providing a far more efficient formula for the dimension of ^K(W) than the one of Theorem <ref>, as well as a basis over K. Let W/K_B denote a set of left coset representatives of K_B in W.

 Let W be an irreducible complex reflection group and K a proper field.
Then


    _K ^K(W)=∑_B∈𝒞^K_adm| W || K_B |,

and the set {we_B| B∈𝒞^K_adm, w∈ W/K_B} is a basis for ^K(W).


 Recall the following formula for the dimension of ^K(W) of Corollary <ref>: 

    _K ^K(W)=∑_B∈𝒞^K_adm| W |/|(B) |·(_K K(B)_K(B)(B))

By Proposition <ref>, the dimension of  K(B)/_K(B)(B) is _K K(B) - _K (K_B)
which, in turn, equals |(B) | /| K_B |. Replacing this in the above formula, gives the dimension 
of the statement. 




Recall, now, the setting of Lemma <ref>, i.e. ^K(W) is the union of the descending chain of ideals I^(K)_r=∑_B∈𝒞, | B |≥ r KWe_B of ^K(W) (Definition <ref>). We show that, for all r∈ℕ, we have 
 

    I^(K)_r=∑_B∈𝒞^K_adm, w∈ W/K_B Kwe_B + I^(K)_r+1.

Induction on r then yields that the set {we_B| B∈𝒞^K_adm, w∈ W/K_B} spans ^K(W). Since its cardinality is equal to _K ^K(W) as this is given above, it is a basis.
 
 To show Equation (<ref>), recall that, by Lemma <ref>, for every B∈𝒞 of cardinality r we have
_KW(B)e_B⊆ I^(K)_r+1.

If B∉𝒞^K_adm, then Corollary <ref> implies that _KW(B)=KW, and, so, KWe_B⊆ I^K_r+1. This already yields that 
I^(K)_r=∑_B∈𝒞^K_adm KWe_B + I^(K)_r+1.




If B∈𝒞^K_adm, then, by Proposition <ref>, we have _K(B)(B)=^χ(K_B) for some 
χ: K_B→ K. Hence, since _K(B)(B)⊆_KW(B) (Definition <ref>), then _KW(B)e_B⊆ I^(K)_r+1 implies that
(h-χ(h))e_B∈ I^(K)_r+1 for all h∈ K_B. Given this, if w_1,w_2 belong to the same coset of K_B, i.e. w_2=w_1h, for
some h∈ K_B, then writing w_2e_B=χ(h)w_1e_B+w_1(h-χ(h))e_B implies that w_2e_B ∈ Kw_1e_B + I^(K)_r+1, which finally yields Equation (<ref>), and concludes the proof.




















 §.§ The infinite series


This subsection contains an analysis of the admissible collections and admissible pairs for the complex reflection groups in the infinite series. The reader may want to consult Subsection <ref> for an overview of the necessary elements we will be using. In all statements, we will assume that K is a proper field for the group in mention. 

We restate here Lemma <ref>. 



In G(m,p,n),


  * if i≠ j, then H_i and H_j are non-transverse and the reflections that map one to the other (in any order) are the reflections (ij)_κ, κ∈ℤ;


  * H_i' and H_ij^κ are transverse if and only if i,j  ≠ i'; in any case, there exist no reflections mapping one to the other;


  * if j_1≠ j_2, then H_ij_1^κ_1 and H_ij_2^κ_2 are non-transverse and the only reflection mapping one to the other (in any order) is (j_1j_2)_κ_2-κ_1;


  * if i_1,j_1 ≠ i_2,j_2, then H_i_1j_1^κ_1 and H_i_2j_2^κ_2 are transverse;



  * for (m,p)≠ (2,2), if  i≠ j, then H_ij^κ_1 and H_ij^κ_2 are not transverse and the reflections mapping the former to the latter are  (ij)_κ, for 2κ≡ κ_1+κ_2  m, and t_i^κ_1-κ_2, t_j^κ_2-κ_1, provided that κ_2-κ_1 ≡ 0  p. 


  * if (m,p)=(2,2), then H_ij and H_ij^1 are transverse.  






A quick way to summarize transversality according to this lemma is the following. In G(m,p,n) for (m,p)≠ (2,2), two hyperplanes expressed as above (notation of Subsection <ref>) are not transverse if and only if they have at least one index in common; in G(2,2,n) two hyperplanes are non-transverse if and only if they have exactly one index in common.




From the above lemma it is also clear that, for n>2, all reflections (ij)_κ belong to the same conjugacy class of G(m,p,n) and so they share a common parameter in the definition of the Brauer-Chen algebra. We will denote this parameter by μ. 


All mentioned transverse collections in this subsection are assumed to be non-empty. One can verify that if B is the empty collection, then _K(B)(B)=_K(B)(K_B)=0.




 Let W=G(m,p,n), and B∈𝒞 be such that | B | >1 and H_i∈ B for some i=1,…, n. Then, there is s∈ℛ such that μ_ss∈(B). In particular, B∉𝒞^K_adm.
 
 Since Η_i, H_j are non-transverse for all i,j (Lemma <ref>), then B can contain at most one such hyperplane. Let α=H_i∈ B and choose a second hyperplane in B, say β which must be of the form H_i'j^κ. Let γ=H_i'i ; then γ⋔̸α,β, and so σ^γ_β,α∈(B).  Again, from Lemma <ref>, we get that σ^γ_β,α=μ(ij)_κ, which implies the statement.






 Let W=G(m,p,n) and B={H_i} for some i. Then, K_B is the pointwise stabilizer W_H_i of H_i, and has order m/p. Moreover, we have 𝒜W(B)=𝒜W(K_B-1).

 
One may check that the definition of K_B gives the pointwise stabilizer of H_i, i.e. the subgroup generated by ℛ_B={ t_i }, which has order m/p.  Furthermore, since B consists of a single hyperplane, then Σ^ H_B={0}, for all H∈ℬ, and, hence (B)=ℛ_B-1, which gives the result.


Lemma <ref> implies that an admissible transverse collection with more than one hyperplane must contain only hyperplanes of the form H_ij^κ. For the groups G(m,p,n) with (m,p)≠ (2,2), by Lemma <ref>, such a collection is transverse if and only if every index i and j as above appears in at most one hyperplane in B. The following proposition shows that all such transverse collections are admissible.


 Let W=G(m,p,n) and B a transverse collection of the form {H_i_1j_1^κ_1,…, H_i_rj_r^κ_r}, with all indices distinct. Then 𝒜W(B)=𝒜W(K_B-1), and, in particular, B∈𝒞^K_adm. Finally, the order of K_B is 
2^rm^r-1r !.
 

Suppose, first,that B={H_i_1j_1,H_i_2j_2,… ,H_i_rj_r}; we show at the end of the proof that the general case is implied under a suitable conjugation. 
Recall that (B)=(ℛ_B-1) ∪ (∪_γ∈ℋΣ^γ_B), where 
Σ^γ_B={σ^γ_α,β|α,β ∈ B, γ⋔̸α,β } (Definition <ref>).

To prove the statement, we show that for every 𝒜W-module V and v∈ V, (B)v=0 is equivalent to (K_B-1)v=0; let V be such a module and v∈ V. The relation (ℛ_B-1)v=0 is equivalent to ((ij)-1)v=0, for all H_ij∈ B.

We turn, now, to the relations Σ^γ_Bv=0.
First, let γ=H_i for some i. By Remark <ref>, two hyperplanes are non-transverse if and only if they share some index. Since all indices that appear in B are distinct, this implies that γ is non-transverse with at most one hyperplane in B, which implies that Σ^γ_B={0}.

Now, let γ=H_ij^κ, for some i,j,κ. By the same observation, we see that if γ is non transverse with more than one hyperplane in B, then the indices i,j appear in two different elements of B. That means that there are α= H_ij', β=H_ji'∈ B, and these are the only hyperplanes of B non-transverse with γ. Lemma <ref> now gives:
σ^γ_α,β=μ(j'j)_κ-μ(i'i)_-κ=μ((j'j)_κ-(ii')_κ), so, σ^γ_α,βv=0 is
((j'j)_κ-(ii')_κ)· v=0, or, equivalently ((ii')_κ(j'j)_κ-1)· v=0. Hence, letting i,j run through all possible distinct indices appearing in B, we get that (∪_γ∈ℋΣ^γ_B)· v=0 is equivalent to 
((ii')_κ(j'j)_κ-1)· v=0, for all H_ij',H_i'j∈ B, and κ∈ℤ.

This shows, already, that (B)· v=0 is equivalent to (K_B^0-1)v=0, where K_B^0 is the group generated by {(ij)| H_ij∈ B}∪{(ii')_κ(j'j)_κ| H_ij',H_i'j∈ B,κ∈ℤ}. We show that this subgroup coincides with K_B. For that, since the set
{(ij)| H_ij∈ B}=ℛ_B is included in both generating sets, it is enough to show that {(ii')_k(j'j)_k| H_ij',H_i'j∈ B,κ∈ℤ} is equal to the set of s_2^-1s_1 where s_1,s_2 satisfy conditions (c1), (c2) of Definition <ref>, i.e.
s_1B=s_2B≠ B and s_1H≠ s_2H for all H∈ B\ s_1B.

For H_ij',H_i'j∈ B, and κ∈ℤ the reflections (ii')_κ, (j'j)_κ fix all hyperplanes of B but H_ij',H_i'j, which they map to H_j'i'^κ,H_ij^κ, and 
H_ij^κ,H_j'i'^κ, respectively, in that order. So, we can see that they verify the conditions (c1), (c2). Hence, K_B^0⊆ K_B.

For the inverse inclusion, let s_1,s_2∈ℛ satisfy (c1), (c2). We exclude certain cases for s_1, s_2. If s_1 is of the form t_i for some i, then one can see that it fixes all but at most one hyperplane of B, in which case conditions (c1), and (c2), cannot be satisfied. This is also the case when s_1 is of the form (ij)_κ with at most one of i,j appearing in B, as well as with their appearing in the same hyperplane. Hence, s_1,s_2 must be of the form (ij)_λ_1,(i'j')_λ_2 with i,j and i',j' appearing, respectively, in different hyperplanes of B. Now, all hyperplanes not containing i,j will be fixed by s_1 and vice versa for s_2; conditions (c1), (c2) imply that s_1,s_2 fix the same hyperplanes of B, and this in turn implies that
H_ij', H_i'j∈ B or H_ii',H_jj'∈ B. Both cases are equivalent, so suppose the second is true.
In that case, since s_1B=s_2B, we must have λ_1=λ_2, which implies that 
s_2^-1s_1∈ K_B^0. Hence K_B=K_B^0, and this shows that (B)v=0 is equivalent to (K_B-1)v=0. As far as the order of K^0_B is concerned we will calculate it in the appendix of this article, where we will also obtain an alternative characterization in terms of matrices.

Now, for the general case, notice that  
{H_i_1j_1^κ_1,… ,H_i_rj_r^κ_r} is the image of B via the element 
∏_i=1^r t̂_j_i^k_i∈ G(m,1,n), where t̂_j_i is the reflection with hyperplane H_j_i in G(m,1,n) acting as multiplication by ζ of the j_i-th coordinate. Now, the action of G(m,1,n) respects transversality, and, also g ·ℛ_H→ H'· g^-1 =ℛ_gH→ gH' for all g∈ G(m,1,n) and H,H'∈ℋ. From this, one can see that g(B)g^-1=(gB), as well as, 
gK_Bg^-1=K_gB, and hence, 𝒜(gB)=𝒜W(K_gB-1), which concludes the proof.




The above proposition already concludes the classification of admissible collections and their corresponding pairs for the groups G(m,p,n), with (m,p)≠ (2,2). It also enables us to obtain a numerical formula for the dimension of the Brauer-Chen algebra in these cases.


 Let W=G(m,p,n), and suppose that (m,p)≠ (2,2). Then a nonempty transverse collection B is K-admissible if and only if it has cardinality 1 or does not contain a hyperplane of the form H_i. In that case, _K(B)(B)=(K_B). Finally, the dimension of ^K(W) is equal to:



    n! m^n/p+(1-δ_pm)n!m^n-1n+m^n+1/p∑_r=1^2r≤ n(n!/r!2^r(n-2r)!)^2(n-2r)!,

where δ_pm is 1 if p=m and 0 otherwise.


 The characterisation of admissible collections is a consequence of the previous results and discussion. Now, Propositions <ref> and <ref> imply that if B∈𝒞^K_adm, then KW(B)=KW(K_B). Proposition <ref> tells us that
_K(B)(B) is equal to π_1(KW(B)), where π_w denotes the projection of KW onto the coset wK(B). Now, by Proposition <ref> the latter is equal to 
π_1(KW(K_B-1)) which is in turn equal to (K_B) since K_B-1 lies inside K(B). This gives the first part of the statement. In particular, we have that _K(K(B)/_K(B))=|(B) |/ | K_B |.


Now, recall, the formula for the dimension from Corollary <ref>:


    ∑_B∈𝒞^K_adm| W |/|(B) |·_K(K(B)_K(B)(B)).


Replacing the above into this formula we get:


    | W | + ∑_B∈𝒞^K_adm, B≠∅| W |/| K_B |.


If p≠ m we have the collections {H_i} with | K_B | =m/p.  
Apart from them, for all collections of the form {H_i_1j_1^κ_1,… H_i_rj_r^κ_r} the group K_B, by Proposition <ref> has order 2^rm^r-1r!, and for each r≤ n/2 the number of such collections is:


    n  2n-2  2…n-2(r-1)  2m^r/r!=n!m^r/2^r(n-2r)!r!.

So the sum ∑_B | W |/| K_B |,
where B runs all such collections of size r is


    n!m^r/2^r(n-2r)!r!| W |/| K_B |=m^n+1/p(n!/r!2^r(n-2r)!)^2(n-2r)!.

Adding it all together, one obtains the expression of the statement.



We turn, now, to the groups G(2,2,n) which only contain hyperplanes of the form H_ij and H^1_ij, since H_ij^κ=H_ij^κ' if κ≡ κ'  2. Remark <ref> implies that such a collection B is transverse if and only if for every H_ij^κ∈ B, i,j do not appear as indices in any other hyperplane of B apart from, possibly, H_ij^κ+1.

 Let W=G(2,2,n) and B a transverse collection such that H_i_1j_1,H_i_1j_1^1,H_i_2j_2^κ∈ B but H_i_2j_2^κ+1∉B, for some i_1,j_1,i_2,j_2. Then, B∉𝒞^K_adm.

 Let V be a K(B)-module. We show that (B)·V̂=0  implies V=0 (recall that V̂ is the image of V in the induced module of KW). For this, take the element σ^γ_α,β∈(B) for α=H_i_1j_1, β=H_i_2j_2^κ and γ=H_i_1j_2, and see that σ^ γ_α,βV̂=0 gives μ((j_1j_2)-(i_1i_2)_κ)V̂=0. Now, we will show that the reflections (j_1j_2) and (i_1i_2)_κ belong to different cosets of (B); hence, μ((j_1j_2)-(i_1i_2)_κ)V̂=0 implies (i_1i_2)_κV̂=(i_1i_2)_κV̂=0 which yields, of course, V=0.


To verify that (j_1j_2),(i_1i_2)_κ belong to different cosets of (B), notice that the only hyperplanes that are not fixed by (j_1j_2),(i_1i_2)_κ are H_i_1j_1,H_i_1j_1^1 and H_i_2j_2^κ, so it suffices to check their respective images via (j_1j_2),(i_1i_2)_κ. For this,
(j_1j_2) maps H_i_1j_1,H_i_1j_1^1,H_i_2j_2^κ to H_i_1j_2,H_i_1j_2^1,H_i_2j_1^κ, and (i_1i_2)_κ maps the same hyperplanes to 
H_i_2j_1^κ,H_i_2j_1^κ+1,H_i_1j_2^2κ. One can see now that (j_1j_2)B≠ (i_1i_2)_κB, since, for example, H_i_1j_2^1 belongs to the first collection but not the second. This concludes the proof. 




 Let W=G(2,2,n) and B a collection of transverse hyperplanes of the form 
{H_i_1j_1,H_i_1j_1^1, … ,H_i_rj_r,H_i_rj_r^1}. Then 𝒜W(B)=𝒜W(K_B-1) and, in particular, B∈𝒞^K_adm. Finally, the order of K_B is 2^r+n-1r!.



 We show, as we did in Proposition <ref>, that for all 𝒜W-modules V and v∈ V, (B)v=0 is equivalent to (K_B-1)v=0. So let V be an 𝒜W-module and v∈ V.

The relations (ℛ_B-1)v=0 give ((ij)-1)v=((ij)_1-1)v=0 for all H_ij∈ B.
Next we see what the relations Σ^γ_Bv=0, γ∈ℋ yield. Let γ=H_ij^κ, for some i,j,κ. By Remark <ref>, two hyperplanes 
of G(2,2,n) are non-transverse if and only if they have exactly one index in common. So, if γ is non-transverse with more than one element of B (otherwise Σ^γ_B={0}), then either hyperplanes a=H_i_1j, b=H_i_1j^1, c=H_ij_1, , d=H_ij_1^1 belong to B, in which case, these are the only hyperplanes γ is non-transverse with, or α=H_ij_1, β=H_ij_1^1∈ B, and j does not appear in any hyperplane of B, in which case α,β are the only hyperplanes of B with which γ is non-transverse.

For the first case, by Lemma <ref>, for each of the hyperplanes a,b,c,d, there is a unique reflection mapping them to γ. These are, respectively, 
r_1=(i_1i)_κ, r_2=(i_1i)_κ+1, r_3=(j_1j)_κ, r_4=(j_1j)_κ+1. So, Σ^γ_Bv=0 yields 6 relations (all the possible differences of the above reflections), which are, equivalent, for example, to
μ(r_1-r_2)v=μ(r_2-r_3)v=μ(r_3-r_4)v=0, or, equivalently (r_2^-1r_1-1)v=(r_3^-1r_2-1)v=(r_4^-1r_3-1)v=0. 

The second case yields the relation σ^γ_α,βv=0 which, by Lemma <ref>, is (u_1-u_2)v=0, or equivalently, (u_2^-1u_1-1)v=0, where
u_1=(j_1j)_κ and u_2=(j_1j)_κ+1.  

Given the above, one can see that letting γ vary, and considering the relations 
(ℛ_B-1)v=0 as well, we obtain a subgroup K_B^0 for which (B)v=0 if and only if (K_B^0-1)v=0. Although it is not necessary for the rest of this proof, we mention here, for later reference, the following explicit description that we obtain for K_B^0:

    K_B^0=⟨ (ij_1)_κ, (i_1i)_κ(i_1i)_κ+1,(i_1i)_κ(j_1j)_κ, (j_1j_2)_κ(j_1j_2)_κ+1⟩,

with H_ij_1, H_i_1j∈ B, κ∈ℤ, and  1≤ j_2 ≤ n. Elements of the first form in the above presentation come from the relations (ℛ_B-1)v=0. Elements of the second and third form come from the relations (r_1r_2^-1-1)v=0 and (r_3r_1^-1-1)v=0, respectively, and the elements of the fourth form from relations (r_3r_4^-1-1)=0 and (u_1u_2^-1-1)v=0, where, for the latter, we have replaced the variable j with j_1 and j_1 with j_2 to give a more concise description. We show that K_B^0 coincides with K_B. 

For the inclusion K_B^0⊆ K_B, since, by the definition of K_B (Definition <ref>), ℛ_B⊆ K_B, it suffices to show that (u_1,u_2) as well as all pairs among the reflections r_1,r_2,r_3,r_4 satisfy conditions (c1), (c2) of the definition of K_B i.e. s_1B=s_2B≠ B, and s_1H≠ s_2H for all H∈ B\ s_1B. 

For u_1=(j_1j)_κ, u_2=(j_1j)_κ+1, with j not appearing in any hyperplane of B, the only hyperplanes of B that are not left invariant by u_1,u_2 are α=H_ij_1,β=H_ij_1^1 which are mapped via u_1,u_2, respectively, to H_ij^κ,
H_ij^κ+1 and H_ij^κ+1, H_ij^κ, in that order. One can see from this that  u_1,u_2 satisfy conditions (c1),(c2).


Similarly, for the pairs of reflections among r_1,r_2,r_3,r_4 computing the images of the quadruple  (a,b,c,d) via these reflections we find, respectively, the following quadruples:
(H_ij^k, H_ij^κ+1, H_i_1j_1^κ, H_i_1j_1^κ+1), (H_ij^κ+1, H_ij^κ, H_i_1j_1^κ+1, H_i_1j_1^κ), (H_i_1j_1^κ, H_i_1j_1^κ+1, H_ij^κ, H_ij^κ+1), (H_i_1j_1^κ+1, H_i_1j_1^κ, H_ij^κ+1, H_ij^κ). Since all hyperplanes of B other than a,b,c,d are fixed by these reflections, one can verify from the above mappings that all pairs (r_l,r_m), for l,m=1, 2, 3, 4 verify (c1), (c2), and, hence, r_m^-1r_l∈ K_B, l,m=1, 2, 3, 4. Now, since K_B^0 is generated by all such products for the different values of γ, together with ℛ_B, we get that K_B^0⊆ K_B.

For the inverse inclusion, since, again, ℛ_B⊆ K_B^0, we show that for every pair of reflections s_1,s_2 satisfying (c1),(c2), we have s_2^-1s_1∈ K_B^0. Observe first, that for every pair of hyperplanes in G(2,2,n), there is at most one reflection mapping one to the other, and all reflections belong to the same orbit (see, for example, Lemma <ref>). We show that for every pair s_1,s_2∈ℛ satisfying (c1), (c2), μ(s_1-s_2)∈ Σ^H_B for some hyperplane H, which implies that s_2^-1s_1∈ K_B^0, since K_B^0 is generated, by definition, by all such elements. For this, 
by (c1), i.e. s_1B=s_2B≠ B there is some hyperplane H'∈ B such that s_1H'≠ H'. 
Take H=s_1H' and H”=s_2^-1H∈ B. Now, by Lemma <ref>, since, 
of course ℛ_H'→ H, ℛ_H”→ H≠∅, we have that
H is non transverse with H',H”, which implies that 
σ^H_Η',Η”∈ Σ^H_B. By the observation in the beginning of this paragraph, we get that ℛ_H'→ H={s_1}, and  ℛ_H”→ H={s_2}, which implies that σ^H_Η',Η”=μ(s_1-s_2). Hence, K^0_B=K_B.
The order of K_B is calculated in the appendix using the group K_B^0.



The above results conclude the classification of admissibility for the groups G(2,2,n). Again, we sum up the situation in the next proposition, giving a numerical expression for the dimension of the Brauer-Chen algebra, as well.




 
 Let W=G(2,2,n). A transverse collection B is K-admissible if and only if whenever H_ij,H_ij^1,H_i'j'^κ∈ B, we have H_i'j'^κ+1∈ B as well. In that case, _K(B)(B)=(K_B). Finally, the dimension of ^K(W) is:


    n!2^n-1+(2^n+1)∑_r=1^2r≤ n(n!/r!2^r(n-2r)!)^2(n-2r)!.



The characterisation of admissible collections of the statement is immediate from the previous results. Notice also that it implies that a transverse collection of G(2,2,n) is admissible if and only if it is of the form {H_i_1j_1^κ_1, … , H_i_rj_r^κ_r}, or {H_i_1j_1,H_i_1j_1^1 … , H_i_rj_r,H^1_i_rj_r}, with all indices distinct in both cases. Transverse collections of both of these types are admissible by Propositions <ref> and <ref>, respectively, and the same propositions imply that if B is of these types, then KW(B)=KW(K_B-1).

Now, exactly as we did in the beginning of the proof of Proposition <ref>, this yields that 
for B∈𝒞^K_adm, _K(B)(B)=(K_B), and the formula for the dimension of ^K(W) of Corollary <ref> becomes:

 
    | W | + ∑_B∈𝒞^K_adm, B≠∅| W |/| K_B |.

 
 
For every r≤ n/2 the number of transverse collections of the form {H_i_1j_1,H_i_1j_1^1 … , H_i_rj_r,H^1_i_rj_r} is: 


    n  2n-2  2…n-2(r-1)  2/r!=n!/2^r(n-2r)!r!,

and the order of K_B, by Proposition <ref>, for each such collection is equal to 2^r+n+1r!. So the sum ∑_B| W | /| K_B |, where B runs over all such collections is: 
 

    ∑_r=1^2r≤ nn!/(n-2r)!2^rr!n!2^n-1/2^r+n-1r!=∑_r=1^2r≤ n(n!/r!2^r(n-2r)!)^2(n-2r)!.


Finally, for the transverse collections of the form {H_i_1j_1^κ_1,… ,H_i_rj_r^κ_r}, the corresponding sum was calculated for Proposition <ref>, and equals:


    2^n∑_r=1^2r≤ n(n!/r!2^r(n-2r)!)^2(n-2r)!.
 
Again, adding it all up (and the order of the group) gives the expression of the statement.


































 §.§ The exceptional groups



In this subsection we describe the classification of admissible collections and the corresponding admissible pairs for the exceptional complex reflection groups. To obtain this classification we used computational methods. After some new definitions, we describe the computational results that lead to the classification and explain the algorithm we used to verify them. A table containing the obtained numerical data concerning the admissible pairs for each exceptional group as well as the dimension of the corresponding Brauer-Chen algebra can be found in the appendix. 



All mentioned transverse collections will be assumed non-empty.




  §.§.§ A convenient description for (B)


Let W be a complex reflection group and B a transverse collection. By Definition <ref>, we have 
    (B)=∪_w∈ Wπ_w((B)),
 where π_w denotes the projection 
KW→ wK(B) with respect to the decomposition KW=⊕_w∈ W/(B)wK(B).
The coset w(B) depending only on the transverse collection B'=wB, we denote, in this subsection, the corresponding projection by π_B', i.e. if x=∑_w∈ Wλ_ww∈ KW, then π_B'(x)=∑_wB=B'λ_ww. With this notation we have
(B)=∪_B'∼ B π_B'((B)). 


Now, recall that 
(B)=(ℛ_B-1)∪ Σ_B, where Σ_B is the set consisting of all elements 
    σ^H_H_1,H_2=∑_s∈ℛ_H_1→ Hμ_ss-∑_s∈ℛ_H_2→ Hμ_ss,

with H_1,H_2∈ B, H∈ℋ, and H⋔̸H_1,H_2.  

For the projections of the set ℛ_B-1, since it lies inside K(B), we have π_B(ℛ_B-1)=ℛ_B-1, and
π_B'(ℛ_B-1)=0 for all B'≠ B. 

Secondly, for all H,H_1,H_2∈ℋ, and B' in the orbit of B, we have


    π_B'(σ^H_H_1,H_2)= ∑_s∈ℛ_H_1 → H, B→ B'μ_ss-∑_s∈ℛ_H_2→ H, B→ B'μ_ss.

If, now, H_1,H_2∈ B, and H⋔̸H_1,H_2, then H∉B; so, for B'=B, we have π_B'(σ^H_H_1,H_2)=0. This is also the case when ℛ_B→ B'=∅, since the above summations are empty. With this in mind, we have the following expression for (B), which we will use later:




    (B)=(ℛ_B-1)∪ (∪_B'∼_ℛB,B'≠ Bπ_B'(Σ_B)),

where B'∼_ℛ B stands for ℛ_B→ B'≠∅.


We define the sets Θ_B^1:={μ_ss | s∈ℛ\ℛ_B} and Θ_B:=Θ_B^1∪ℛ_B∪{1},
and let M_B denote the free ℚ-module on Θ_B. Also, let M_B^1, M_B^2 be, respectively, the submodules ℚΘ_B^1 and ℚℛ_B∪{1} of M_B.



From the above, one can see that we can consider (B) as a subset of M_B for which

    (B)=((B)∩ M_B^1 )∪ ((B)∩ M_B^2),
 and 
    (B)∩ M_B^2=ℛ_B-1.



 Let 

    θ:   ℛ∪{1}→ Θ_B;
        s↦ μ_ss,  if s∈ℛ\ℛ_B,
        s↦ s,  if s∈ℛ_B∪{1}.

We define the following sets:

    D_B:={θ(s_1)-θ(s_2)| s_1,s_2∈ℛ}∩ℚ(B)⊆ M_B,
       P_B:={(s_1,s_2)| s_1,s_2 ∈ℛ\ℛ_B, μ_s_1s_1-μ_s_2s_2∈ D_B}⊆ℛ^2,
       D_B^0:={s_2^-1s_1-μ_s_2μ_s_1^-1| (s_1,s_2)∈ P_B}∪ (ℛ_B-1)⊆𝒜W.










 Let s_1,s_2∈ℛ∪{1}. If θ(s_1)-θ(s_2)∈ℚ(B), then either s_1,s_2∈ℛ_B∪{1}, and, hence, θ(s_1)-θ(s_2)=s_1-s_2, or s_1,s_2∈ℛ\ℛ_B, and, hence,
θ(s_1)-θ(s_2)=μ_s_1s_1-μ_s_2s_2. 

 Suppose that 
θ(s_1)-θ(s_2)∈ℚ(B) and that s_1,s_2 do not satisfy the condition of the statement. Let
s_1∈ℛ\ℛ_B and s_2∈ℛ_B∪{1}.  Hence, θ(s_1)∈ M_B^1 and θ(s_2)∈ M_B^2.  Since ℚ(B) respects the decomposition M_B=M_B^1⊕ M_B^2,
i.e. (B)=(ℚ(B)∩ M_B^1)⊕ (ℚ(B) ∩ M_B^1), then the projection of θ(s_1)-θ(s_2) onto M_B^2 belongs to (B). Hence, we get θ(s_2)=s_2 ∈(B)∩ M_B^2=ℚ(ℛ_B-1), and one can see that this is impossible.



 We have 

    D_B={μ_s_1s_1-μ_s_2s_2| (s_1,s_2)∈ P_B}∪{s_1-s_2| s_1,s_2∈ℛ_B∪{ 1 }}.
 In particular,  𝒜W D_B=𝒜W D_B^0.






















  §.§.§ The main points of the algorithm


We explain now the main points of the algorithm we used to obtain the results for the admissibility for the exceptional complex reflection groups. Almost all created lists will be used again in the next section, where we study the freeness of the Brauer-Chen algebra.



  
The main core.
We consider the following data. A complex reflection group , given as an abstract group (a group of permutations, for example),
and a list  of the elements of  that correspond to the distinguished reflections (Subsection <ref>). Let, also,  
be the field of definition of  and  be a list containing a root for each distinguished reflection. 

The elements of  are in bijection with the hyperplanes of ; let H_i be the hyperplane that corresponds to the
distinguished reflection in position i of . This way, testing, for an element w∈, whether 
wH_i=H_j, amounts to testing whether w[i]w^-1=[j]. Also, by Lemma <ref>, checking that H_i⋔ H_j is
equivalent to checking that there is not any root in  lying in the -span of {}
apart from these two.

It is convenient to create a table containing all the necessary relations of transversality and conjugation between the hyperplanes
of . For that, let  be a list containing all reflections of  (take, for example, all non-trivial powers of the 
elements of ), and create a table  as follows: if H_i⋔ H_j, then ; otherwise,
 is a list with the positions of the elements of  that map H_i to H_j. 

Using the table , we can find all transverse collections of , and, using standard orbit algorithms, we can
obtain a representative for each orbit of transverse collections. We represent a transverse collection as a list of positions in
.




  
Lists for (B), D_B, P_B. Now, let  be such a list representing a transverse collection B. Let N be the cardinality of ℛ. We create a list 
 of vectors in  ℤ^N+1 representing the set (B) with respect to the basis Θ_B of M_B.

First, we make a list  containing the positions in  
of all non-trivial powers of the elements , for i∈; this list represents the set ℛ_B.
Also, we make a list  which contains the lists representing the images of B under all reflections; this represents the set of all B'∼_ℛB.  


Let v(i,n) denote the vector of length n, with 1 in position i and 0 elsewhere. First, for all
i∈ we add the vector v(i,N+1)-v(N+1,N+1) to . This vector represents the element 
-1 with respect to Θ_B. In this way, we include all vectors  representing ℛ_B-1 to our list.

Next, for all B'∼_ℛB, B'≠ B we add the vectors representing the set π_B'(Σ_B), which (apart from 0) consists of the 
elements π_B'(σ^H_H_1,H_2) for H_1,H_2∈ B, H∈ B' and H⋔̸B. Instead of imposing controls on all candidate triples of hyperplanes, going back to Lemma <ref>, one may see that,
since ℛ_B→ B'≠∅, choosing H∈ B' and H_1,H_2∈ B non-transverse with H amounts to 
choosing H∈ B'\ B and H_1,H_2∈ B\ B'. 

With this in mind, for every  in , we first
create a table , with rows and columns corresponding to the elements of \ and \, respectively, and
whose (i,j)-cell is a list of positions of all reflections in  that map H_ to H_ and B to B' (note that, in this way, for the indices to agree,
one has to make sure that the intersection of , takes up the last spots on each list). Now, for all 
i,j,k in the range of , we add the vector 
∑_l∈v(l,N+1)-∑_l∈v(l,N+1) to the list , as
well. This vector represents the element π_B'(σ_H',H”^H) where H,H',H” are the hyperplanes H_,H_,H_, respectively. In this way we include
all vectors representing π_B'(Σ_B) to , as well. 

Before we state our first computational result, let  be a list of all vectors v(i,N+1)-v(j,N+1), for i,j=1,… N+1 that lie in 
the ℚ-span of ; this represents the set D_B with respect to the basis Θ_B. Also, let
 be the list of all pairs (i,j), i,j=1,… ,N, with i,j∉, such that v(i,N+1)-v(j,N+1)∈; this 
represents the set P_B. 



First Computational Result. Let W be an exceptional complex reflection group and B a transverse
collection. Then



  (A1)θ(s)∈(B),  for some s∈ℛ∪{1}, or

  (A2) 


  a. ℚ(B)=ℚD_B, and

  b. ⟨ s_2^-1s_1, s | (s_1,s_2)∈ P_B, s∈ℛ_B⟩ = K_B

 


The verification of the above result for a given transverse collection represented by  is straightforward using the previously constructed lists. Checking, in particular, the first condition amounts to checking whether  contains the vector u(i,N+1) for some i.

 Let W be an exceptional complex reflection group, K a proper field, and B a transverse collection. Notice, first of all, that the
two properties (A1), (A2) above are mutually exclusive, since a vector θ(s) cannot be a linear combination of vectors of the form θ(s_1)-θ(s_2), which comprise D_B.

If B satisfies (A1), then, since θ(s) is invertible, B is not
K-admissible. 

Suppose now that B satisfies (A2). By (A2.a), ℚ(B)=ℚD_B, and, thus, admissibility of a pair
(B,V) is equivalent to D_B·V̂=0, which is, is turn, equivalent to D_B^0·V̂=0, as Corollary <ref> implies.
Since K_B⊆(B), (A2.b) implies 
that s_2^-1s_1∈(B) for all (s_1,s_2)∈ P_B, which, in turn, yields that 
D_B^0⊆ K(B). So we can drop the hat on V̂, and write D_B^0· V=0. This already implies, since, for any K(B)-module V, the pair (B,V) is K-admissible if and only if _KStab(B)(B)· V=0, that _K(B)(B) is equal to (D_B^0), where (D_B^0) denotes the two-sided ideal of K(B) generated by D_B^0.

Now, if μ_s_1=μ_s_2 for all (s_1,s_2)∈ P_B, then D_B^0={s_2^-1s_1-1| (s_1,s_2)∈ P_B}∪ (ℛ_B-1), and, in combination with (A2.b), we get that (B,V) is admissible if and only if 
(K_B-1)V=0; in other words, (D_B^0)=(K_B). This is especially the case if s_1,s_2 are conjugate for all (s_1,s_2)∈ P_B. 


Let W be an exceptional complex reflection group and B a transverse collection. If B satisfies (A2) and P_B contains pairs (s_1,s_2) with s_1≁s_2, we say that B is of conditional type. Denote by 𝒞_cond the set of collections of conditional type.



The discussion preceding the above definition yields the next lemma.

 Let W be an exceptional complex reflection group and B∈𝒞\𝒞_cond. For any proper field K, B∈𝒞^K_adm if and only if B does not satisfy (A1), in which case _K(B)(B)=(K_B).


Verifying whether a transverse collection is of conditional type is straightforward
using the list .




Second Computational Result. Let W be an exceptional complex reflection group other than G_25,G_32. Then
W does not have any transverse collections of conditional type. 


We, thus, obtain the following result.
 Let W be an exceptional complex reflection group other than G_25,G_32 and K a proper field. Then B∈𝒞^K_adm if and only if (B) does not contain an element of Θ_B. If B∈𝒞^K_adm, then _K(B)(B)=(K_B).




For the groups G_25 and G_32, there exist transverse collections of conditional type. Unlike collections of non-conditional type, admissibility of such collections depends, in general, on the field of definition. The following general algebraic fact determines the admissibility of these collections for a large number of cases. 

  Let L be a field and G a group. Let l∈ L and g be an element of G of finite order n. Then, if l^n≠ 1, the element g-l∈ LG is invertible.

 Set x=g-l. Since, g^n=1 and g=x+l, then (x+l)^n=1⇔ x^n+nx^n-1l+ … +nxl^n-1+l^n=1⇔ x(x^n-1l+… +nl^n-1)=1-l^n. Now, if 1-l^n≠ 0, then 1-l^n is invertible, and so is x.




Third Computational Result, first part. Let B be a transverse collection of G_25 (resp. G_32). Then
B satisfies satisfies (A2) of the First Computational Result and is of conditional type if and only if | B | =2 (resp. 2 or 3). In that case, for all (s_1,s_2)∈ P_B,
with s_1≁s_2, the order of the element s_2^-1s_1 is 6. 


Let W be the group G_25 or G_32, and K a proper field for W. Then W contains two orbits of reflections, one containing all distinguished reflections, and the other one their inverses. Let μ_1,μ_2 be their respective parameters and set μ:=μ_1/μ_2. 

Let B be a collection of conditional type, i.e., P_B contains some (s_1,s_2) such that s_1≁s_2. Recall that since, by definition, B satisfies (A2) of the First Computational Result, we have that _K(B)(B)=(D_B^0), and the latter ideal is generated by (ℛ_B-1)∪{s_2^-1s_1-μ_s_2μ_s_1^-1| (s_1,s_2)∈ P_B}. By (A2.b), ℛ_B∪{s_2^-1s_1| (s_1,s_2)∈ P_B} generates K_B, so, if (D_B^0)≠ K(B), then mapping ℛ_B↦ 1, and s_2^-1s_1↦ μ_s_2μ_s_1^-1 induces a group homomorphism χ:K_B→𝕌_6 ∩ K, and (D_B^0)=^χ(K_B).

Now, by the Third Computational Result above and its preceding lemma (<ref>), if μ^6≠ 1 in K, then the element s_2^-1s_1-μ_s_2μ_s_1^-1 is invertible, since
μ_s_2μ_s_1^-1∈{μ,μ^-1}; hence, B∉𝒞^K_adm. If μ^6=1 in K, to determine the admissible pairs for B, one may resort - as we did - to computing the ideal (D_B^0).



  
Computing (D_B^0) for G_25,G_32.
Let  be the group G_25 or G_32, and  a list representing a transverse collection B of conditional type (the conditionality property is actually irrelevant for the algorithm).
Let μ∈𝕌_6. Since D_B^0⊂ℚ[μ](B), it suffices to calculate the ideal of ℚ[μ](B) generated by D_B^0. Let  be a list containing the elements of (B) and let N' be its length; suppose, also, that 1 is in the first spot of this list. Finally, let  be the lists containing the positions of reflections in  that make up the two orbits.

We create a list  containing the vectors that represent D_B^0 with respect to the basis . 
In order to do this, for all i∈ we add the vector v(,N')-v(1,1) to the list ; this is the vector representing -1. In this way we include all vectors representing ℛ_B-1. Next, for every (i,j)∈:




  * if i,j∈ or i,j ∈, then we add the vector 
v(^-1,,N'))-v(1,1), representing the element ^-1-1,


  * if i∈ and j∈, then we add the vector
v(^-1,,N'))-μ^-1· v(1,1), representing the element ^-1-μ^-1


  * if i∈ and j∈, then we add the vector 
v(^-1,,N'))-μ· v(1,1), representing the element ^-1-μ

This gives the desired list .


We give, now, ths algorithm that we used to obtain the subspace of ℚ[μ]^N' that corresponds to the ideal (D_B^0). Let  be a list of generators for (B). 
For every element g∈ we produce matrices (g), (g), which are, respectively, the matrices of left and right multiplication by  on ℚ[μ](B), given in the basis . These are the monomial matrices that correspond to the permutations of  induced by multiplying each element by g∈ on the left and right, respectively.
 
Now, let  be a list of vectors of ℚ[μ]^N' representing a set of elements L of ℚ[μ](B). To calculate the subspace of ℚ[μ]^N' that corresponds to the ideal generated by L, we start with the list , and, in step i, we obtain a  basis  of the ℚ[μ]-span of {v,(g)· v,(g)· v  | v∈, g∈}. The list  corresponds to a basis of the vector space spanned by L_i-1∪ (· L_i-1)∪ (L_i-1·). We stop when the lengths of  and  are equal; this is the dimension of the ideal generated by L, and  represents a basis of that ideal. 

We now state the last computational result for this section.




Third Computational Result, second part. Let W be the group G_25 or G_32 and K a proper field. If μ^6=1 in K, and B∈𝒞_cond, then _K(D_B^0)=|(B) | - |(B) |/ | K_B |. In particular, B is K-admissible.




Finally, we have the following result for G_25 and G_32.

 Let W be the group G_25 (resp. G_32) and K a proper field. Let B a transverse collection. 



  * If | B |≠ 2 (resp. 2,3), then B∈𝒞^K_adm, and _K(B)(B)=(K_B) 


  * If | B | = 2 (resp. 2 or 3) and μ^6≠ 1 in K, then B∉𝒞^K_adm. 


  * If | B | = 2 (resp. 2 or 3) and μ^6=1 in K, then B∈𝒞^K_adm and there is a homomorphism χ:K_B→𝕌_6 ∩ K such that _K(B)(B)=^χ(K_B). Furthermore, in that case, _K _K(B)(B)=_K (K_B).



 The first case is a consequence of Lemma <ref> and the first part of the Third Computational Result. The discussion following the latter also implies the second case, and, in combination with the second part of the Third Computational Result, the third case, as well.






§ FREENESS OF (W)











In this section we study for which irreducible complex reflection groups the Brauer-Chen algebra is free over its ring of definition 𝒜. For the cases where the algebra is indeed free, our method will, naturally, be to show that the basis obtained in Theorem <ref> for ^K(W) carries over to the ring of definition.





 §.§ A useful lemma



The following lemma will be critical for the freeness property of (W) in both the infinite series and the exceptional groups case so we mention it here. It is a substitute of Lemma <ref> that works over a ring. Recall the definition of the ideals I_r=∑_B∈𝒞, | B |≥ r e_B𝒜W (Definition <ref>) of (W). Also, let
π_w denote the projection 𝒜W→ w𝒜(B), with respect to the decomposition 𝒜W=⊕_w∈ W/(B)w𝒜(B).

 Let W be a complex reflection group and R an 𝒜-algebra. Let also B be a transverse collection of cardinality r and x=∑_w∈ Wλ_ww∈ RW be such that xe_B∈ I^(R)_r+1. If ∪_λ_w≠ 0wB is a transverse collection, then π_w(x)e_B∈ I^(R)_r+1 for all w∈ W.

 Consider the quotient ^R_r(W)=^R(W)/I^(R)_r+1. In ^R_r(W) we have ∑_w∈ Wλ_wwe_B=0; we will show that π_w(x)e_B=0 in ^R_r(W) for all w∈ W. Pick some w_0∈ W. If π_w_0(x)=∑_w∈ w_0(B)λ_wwe_B ≠ 0 in ^R(W), then we can assume that λ_w_0≠ 0 (otherwise we pick some other w∈ w_0(B) such that λ_w≠ 0). We write ∑_w∈ Wλ_wwe_B=0 as ∑_w∈ w_0(B)λ_wwe_B=-∑_w∉w_0(B)λ_wwe_B, or, equivalently, 


    ∑_w∈ w_0(B)λ_we_w_0Bw=-∑_w∉w_0(B)λ_we_wBw.
 Multiplying both sides of the last equality by e_w_0B on the left, yields
 

    ∑_w∈ w_0(B)δ^rλ_we_w_0Bw=-∑_w∉w_0(B)λ_we_w_0Be_wBw.
 
Now, for the right hand side, if λ_w≠ 0, then w_0B∪ wB is a transverse collection by assumption. Furthermore, if w∉w_0(B), then w_0B≠ wB and so, | w_0B∪ wB | >r. Hence, e_w_0Be_wB=0 in ^R_r(W). So, we have ∑_w∈ w_0(B)δ^rλ_we_w_0Bw=0 or, equivalently, since δ is invertible in R, ∑_w∈ w_0(B)λ_wwe_B=0⇔ π_w_0(x)e_B=0, which implies the statement.



Let W be a complex reflection group and R an 𝒜-algebra. If an element x=∑_w∈ Wλ_ww ∈ RW satisfies ∪_λ_w≠ 0wB∈𝒞, we say that it satisfies the bar condition for B.



The following result is a straightforward corollary of Lemma <ref>.

 Let W be a complex reflection group, R an 𝒜-algebra and B a transverse collection of cardinality r. If every element of (B) satisfies the bar condition with respect to B, then (B)· e_B⊆ I^(R)_r+1.
























 §.§ Freeness in the infinite series


We already have all necessary elements to prove the following result concerning the freeness of (W) for the irreducible complex reflection groups of the infinite series.

 Let W=G(m,p,n). The algebra (W) is a free 𝒜-module with basis {we_B | B ∈𝒞^K_adm, w∈ W/K_B}, for any proper field K.

 Let K be a proper field for W. Notice first that, the set of the statement is a basis for ^K(W) by Theorem <ref>. Hence, it suffices to show that it spans (W) to obtain the result. Our method will be the same as in the proof of Theorem <ref>. We remind it here.

Recall the setting of Lemma <ref>, i.e. that (W) is the union of the descending chain of ideals I_r=∑_B∈𝒞, | B |≥ r𝒜We_B of (W) (Definition <ref>). We show that for all r∈ℕ, we have 
 

    I_r=∑_B∈𝒞^K_adm, w∈ W/K_B𝒜we_B + I_r+1.

Induction on r then yields that the set {we_B| B∈𝒞^K_adm, w∈ W/K_B} spans (W), which implies the result.







Let B be a transverse collection of cardinality r.

Suppose that B∉𝒞^K_adm. We first consider the case (m,p)≠ (2,2). By Proposition <ref>, B is of cardinality at least two and it contains some H_i, in which case, Lemma <ref> gives that, for some s∈ℛ we have 
μ_ss∈(B). Since (B)e_B=0 and μ_ss is invertible in 𝒜, this yields e_B=0. Hence, if (m,p)≠ (2,2), for all r∈ℕ, we have, in particular:


    I_r=∑_B∈𝒞^K_adm𝒜We_B + I_r+1.



We show the same equality for (m,p)=(2,2). By Proposition <ref>, if B∉𝒞^K_adm, there are i_1,j_1,i_2,j_2 and κ such that
H_i_1j_1^0,H_i_1j_1^1,H_i_2j_2^κ∈ B but H_i_1j_1^κ+1∉B.
Taking the element σ^γ_α,β∈(B) for α=H_i_1j_1, β=H_i_2j_2^κ and γ=H_i_1j_2, gives that ((j_1j_2)-(i_1i_2)_κ)e_B=0. We verify that this element satisfies the bar condition (see Definition <ref>), i.e. (j_1j_2)B∪ (i_1i_2)_κB is a transverse collection. For this, notice that, if i,j∉{i_1,i_2,j_1,j_2}, then H_ij,H_ij^1 are fixed by (j_1j_2),(i_1i_2)_κ. Hence, the only hyperplanes in B that are not fixed by (j_1j_2),(i_1i_2)_κ are H_i_1j_1,H_i_1j_1^1 and H_i_2j_2^κ, and it suffices to check transversality of their respective images, i.e. that (j_1j_2)B∪ (i_1i_2)_κB is a transverse collection. We have that 
(j_1j_2) maps H_i_1j_1,H_i_1j_1^1,H_i_2j_2^κ to H_i_1j_2,H_i_1j_2^1,H_i_2j_1^κ respectively, and (i_1i_2)_κ maps the same hyperplanes to 
H_i_2j_1^κ,H_i_2j_1^κ+1,H_i_1j_2^2κ. One can now see that (j_1j_2)B∪ (i_1i_2)_κB is a transverse collection, using, for example Remark <ref>. Hence,
(j_1j_2)-(i_1i_2)_κ satisfies the bar condition. By Lemma <ref>, this implies that π_w((j_1j_2)-(i_1i_2)_κ)e_B∈ I_r+1, for all w∈ W. As one can see from the above calculations, (j_1j_2)B≠ (i_1i_2)_κB, hence (j_1j_2),(i_1i_2)_κ belong to defferent cosets of (B). So, for w=(j_1j_2) for example, we have
π_w((j_1j_2)-(i_1i_2)_κ)=(j_1j_2) which implies that (j_1j_2)e_B∈ I_r, or, since (j_1j_2) is invertible, that e_B∈ I_r+1. This yields Equations (<ref>) above, for the case (m,p)=(2,2) as well.








Suppose now that B∈𝒞^K_adm. If (m,p)≠ (2,2), then by Proposition <ref>, B={H_i} for some i or B={H_i_1j_1^k_1,… , H_i_rj_r^k_r} with i_1,j_1,… i_i,j_r pairwise distinct. In both cases, by Propositions <ref> and <ref>, respectively, we have that (K_B-1)e_B=0. If (m,p)=(2,2) we obtain the same result by Propositions <ref> and <ref>. This implies that if w_1K_B=w_2K_B, then 
w_1e_B=w_2e_B. In particular w_1e_B - w_2e_B ∈ I_r+1, which, together with Equations (<ref>) above, yields Equations (<ref>), and concludes the proof.


 Note that  by Propositions <ref> and <ref>, for the groups in the infinite series admissibility of a transverse collection is, in fact, independend of the field K, and thus, so is the basis for (W) provided by the above proposition.






















 §.§ Freeness for the exceptional groups


We turn now to the study of the freeness property for (W) for the exceptional complex reflection groups. As for the corresponding study of admissibility in Section <ref>, we used computational methods to obtain the results that lead to the determination of the freeness property of (W) for the exceptional complex reflection groups. Again, we explain these results as well as the algorithm used to verify them. Since the case of the group G_26 presented several peculiarities compared to the other exceptional groups, we do not include the computational methods for its treatement, but rather give a self-contained exposition using the description from Example <ref> of Subsection <ref>.







  §.§.§ Study of freeness for W≠ G_26




Let W be a complex reflection group and B a transverse collection. 

 


A hyperplane H is acceptable (with respect to B), if it satisfies the following properties: 


  (a1) H∉B,

  (a2) H⋔̸B,

  (a3) for every H'∈ B non-transverse with H, ∪_s∈ℛ_H'→ HsB is a transverse collection in the orbit of B, i.e. all s∈ℛ_H'→ H map B to the same transverse collection.

A pair of acceptable hyperplanes (H',H”) will be said acceptable if H'≠ H” and there is B'∼_ℛB (that is, with ℛ_B→ B'≠∅) that contains H',H”.



For H,H',H”∈ℋ we define the element 

    τ^H',H”_H:=∑_s∈ℛ_H→ H' μ_ss-∑_s∈ℛ_H→ H” μ_ss∈𝒜W.
 


 Let (H',H”) be an acceptable pair, and H∈ B be non-transverse with both H' and H”. Then, τ^H',H”_H e_B=0.

 Since (H',H”) is an acceptable pair, there is B'∼_ℛ B containing H',H”. Let r∈ℛ_B→ B'. Notice, first of all, that H',H” are transverse, and let H_1=r^-1H'∈ B and H_2=r^-1H”∈ B. This implies, in particular that the pairs H_1, H' and H_2,H” are non-transverse (Lemma <ref>). Since H' is acceptable, by definition, for all reflections s∈ℛ_H_1→ H' we have sB=B'. So, e_H”e_H'e_B=e_H”(∑_s∈ℛ_H_1→ H'μ_ss)e_B=δ· (∑_s∈ℛ_H_1→ H'μ_ss)e_B=δe_H'e_B. In the same way, we get that e_H'e_H”e_B=δe_H”e_B. Now, since H'⋔ H”, e_H',e_H” commute; thus, δe_H'e_B=δe_H”e_B whence (e_H'-e_H”)e_B=0. Since H∈ B is non-transverse with both H',H”, this last equality yields (∑_s∈ℛ_H→ H'μ_ss - ∑_s∈ℛ_H→ H”μ_ss)e_B=0, or, equivalently, τ^H',H”_He_B=0.





We state now the computational result of this section. Its verification will be discussed in the end of this subsection. Let _τ(B) be the set of all τ^H',H”_H, where (H',H”) is an acceptable pair with respect to B and H is non-transverse with both H',H”. For the rest of this subsection, the reader may want to recall the definition of Θ_B, θ_B, D_B, D_B^0 found in Subsection <ref>.







Fourth Computational Result.
Let W be an exceptional group other than G_26 and B a transverse collection. Then 


  (F1) θ(s)∈(B) for some s∈ℛ∪{1}, or

  (F2)
 

  (a) every x∈(B) satisfies the bar condition, and

  (b) if B satisfies property (A2) of the First Computational Result, then D_B⊆ℤ((B)∪_τ(B))










 Let W be an exceptional complex reflection group other than G_26 and R a proper ring for W, with K=(R). The set {we_B | B∈𝒞^K_adm∪𝒞_cond, w∈ W/K_B} spans ^R(W).

 The proof follows our standard method for results of this kind. We show that for all i∈ℕ, we have


    I^(R)_r= ∑_B∈𝒞^K_adm∪𝒞_cond, w∈ W/ K_BRwe_B+I^(R)_r+1,

and the result follows by induction on r. We will use the First Computational Result of the previous section as well.

Let B ∈𝒞\( 𝒞_adm^K∪𝒞_cond). By Lemma <ref>, B satisfies condition (A1) of the First Computational Result, i.e., (B) contains θ(s) for some s∈ℛ∪{1}. If B satisfies (F1) above, then there is s'∈ℛ∪{1} such that θ(s')∈(B), which implies that θ(s')e_B=0 or, equivalently e_B=0, since θ(s') is invertible. If B satisfies (F2), then Corollary <ref> implies that (B)· e_B⊆ I^(R)_r+1. By (A1), (B) contains some θ(s), and this implies that e_B∈ I^(R)_r+1. This yields that for all r∈ℕ,


    I^(R)_r= ∑_B∈𝒞^K_adm∪𝒞_condRWe_B+I^(R)_r+1,


Suppose now that B∈𝒞^K_adm∪𝒞_cond. Again, by Lemma <ref> and the First Computational Result, B satisfies condition (A2). Then B does not satisfy (F1), which would imply (A1); hence, B satisfies (F2), which implies that
(B) e_B⊆ I^(R)_r+1. Now, by (F2.b), D_B⊆ℤ ((B)∪_τ(B)), and since, by Lemma <ref>, _τ(B)e_B=0, then 
we have that D_B· e_B⊂ I^(R)_r+1, which is equivalent to D_B^0· e_B⊂ I^(R)_r+1 (see Corollary <ref>). Recall that 


    D_B^0={s_2^-1s_1-μ_s_2μ_s_1^-1| (s_1,s_2)∈ P_B}∪ (ℛ_B-1),

and, by (A2.b), the set {s_2^-1s_1| (s_1,s_2)∈ P_B}∪ℛ_B generates K_B. This implies that for every h∈ K_B, there is some invertible l_h∈𝒜 (in fact, some product of the parameters) such that (h-l_h)e_B∈ I^(R)_r+1.
Now, if w_1,w_2 belong to the same left coset of K_B in W, i.e. w_2=w_1h for some h∈ K_B, then w_2e_B=w_1he_B=w_1l_he_B+w_1(h-l_h)e_B∈ w_1e_B + I^(R)_r+1. This, together with Equations (<ref>), yields Equations (<ref>), and concludes the proof.






 Let W be an exceptional complex reflection group other than G_25,G_26,G_32, and let K be any proper field. Then, (W) is a free 𝒜-module, with basis 
{we_B| B∈𝒞^K_adm, w∈ W/K_B}. 

 By the previous proposition, (W) is spanned by 
{we_B | B∈𝒞^K_adm∪𝒞_cond, w∈ W/K_B}. 
By the Second Computational Result (Section <ref>), if W is different from G_25,G_32, we have that 𝒞_cond=∅. Hence, the above spanning set is, in fact, {we_B| B∈𝒞^K_adm, w∈ W/K_B}, which, by Theorem <ref>, is a basis for ^K(W), and this implies that it is a basis for (W) as well.


 As in the case of the infinite series, by Proposition <ref>, for an exceptional group other than G_25, G_32, admissibility of a transverse collection is independend of the field K, and so is the basis provided by the above corollary.






 Let W be the group G_25 or G_32.



  * The algebra (W) is not a free 𝒜-module. 

  * If R is a proper ring where μ^6=1 (recall that μ is the quotient of the parameters corresponding to the two orbits of reflections in W), then the algebra ^R(W) is a free R-module, with basis {we_B| B∈𝒞^K_adm, w∈ W/K_B}, where K=(R).




 We show the second statement first. If μ^6=1 in R, then the second part of the Third Computational Result says that 𝒞_cond⊆𝒞^K_adm. So, the spanning set for ^R(W) of Proposition <ref> above coincides with the set of the statement. By Theorem <ref>, the latter is a basis of ^K(W), and, hence, it is a basis of ^R(W) as well. 

Now, by Theorem <ref> again, for any proper field K, the set {we_B| B∈𝒞^K_adm, w∈ W/K_B} is a basis for ^K(W). Let K,K' be proper fields such that μ^6≠ 1 in K but μ^6=1 in K' (see Example <ref> and Remark <ref> to verify the existence of such proper fields). Then, by Proposition <ref>, we have that 𝒞^K_adm⊂𝒞^K'_adm, and, hence, _K'^K'(W)> _K^K(W). This implies that the algebra is not free over 𝒜.








  
How to verify the fourth computational result.

We discuss here the main points of the algorithm we used to verify the computational result of this section. The main core of the algorithm used is established in Subsection <ref>. We remind here the necessary data.

Let  be a complex reflection group with list of reflections and distinguished reflections  and , respectively. We have a table  with rows and columns corresponding to hyperplanes of W such that  if H_i⋔ H_j, or, otherwise,  is a list of positions of reflections in  mapping H_i to H_j. Let  stand for a list representing a transverse collection B of . The list  contains the positions of reflections in  with reflecting hyperplanes in B. Also, the list  contains the vectors in ℤ^N+1, where N=|ℛ|, representing D_B with respect to the basis Θ_B={μ_ss | s∈ℛ\ℛ_B}∪ℛ_B∪{1}.


Similarly to what we did in the previous section for the set (B), representing it with the list , we first construct two lists,  and  of vectors in ℤ^N, representing the sets (B) and _τ(B), respectively, with respect to the basis Θ_B. 

For the list , we add first, for all i∈, the vector v(i,N+1)-v(N+1,N+1) to , including, in this way, all vectors representing the set ℛ_B-1. Next, for all k=1,… , N+1 and i,j∈ such that H_i,H_j⋔̸H_k (we can test this using the table ) we add the vector 
∑_l∈ v(l,N+1)-∑_l∈v(j,N+1) to ; this vector represents the element σ^H_k_H_i,H_j. In this way we include all vectors representing the elements of Σ_B, which gives the desired list. 

Now, using the list , verifying condition 
(F1) amounts to checking whether there is i=1… , N+1 such that
v(i,N+1)∈. For condition (F2.a), verifying the bar condition for an element of 
ℤΘ_B, represented by a vector in ℤ^N+1, is straightforward using the conjugation of distinguished reflections and the transversality information of table .



For the list , we first construct a list of the pairs (i,j) corresponding to acceptable pairs. This is straightforward; all necessary relations for transversality and conjugation are contained in the table . For the acceptable pairs, we may use the list  which contains the lists representing all B'∼_ℛB. Given this, for all (i,j) corresponding to acceptable pairs, and k∈ such that 
H_i,H_k⋔̸H_k we add to  the vector 
∑_l∈v(l,N+1)- ∑_l∈v(l,N+1), which represents the element τ^H_i,H_j_H_k. This completes the construction of this list. 

Now, let  be the concatenation of  and . Condition (F2.b) is equivalent to ⊆ℤ. We conclude with the algorithm that we used to check for linear dependence over ℤ.

Suppose we have vectors l_1,… , l_m, v ∈ℤ^n. To test whether v belongs to the ℤ-span of l_1,… , l_m, we use standard existing algorithms to obtain the information concerning the Smith Normal Form of a matrix. Specifically, let A be the n× m matrix with columns l_1,… ,l_m. Then, the ℤ-span of l_1,… ,l_m is equal to ℤ^m · A, i.e.  the image of ℤ^m under right multiplication by A. The aforementioned algorithms for the Smith Normal Form of A provide us with invertible m× m and n× n, respectively, integer matrices, S and T, and D=_m× n(a_1,… ,a_r), such that SAT=D. Thus, v∈ℤ^m· A is equivalent to v∈ℤ^m· S^-1DT^-1, which is, in turn, equivalent to vT ∈ℤ^m· D, since S^-1 induces an isomorphism ℤ^m→ℤ^m. Now, we can see that, to test if vT ∈ℤ^m · D, we need to check that (vT)_i=0 for i> r and that a_i|(vT)_i for 1≤ 1 ≤ r, where (vT)_i is the i-th coordinate of vT. This concludes the algorithm.












  §.§.§ Freeness for (G_26) 


We prove here that the algebra (G_26) is free.
For this, we remind the description of the irreducible complex reflection group G_26 of Example <ref>.

Let z_1,z_2,z_3 denote the standard coordinates of ℂ^3 and ζ=e^2πi/3. Assume that ℂ^3 is equiped with the standard inner product. The group G_26 is the subgroup of (ℂ^3), generated by the following 3 types of distinguished unitary reflections:



  * t_i, i=1,2,3, with reflecting hyperplanes H_i with equation z_i=0, and order 3,

  * t_κ,λ, κ,λ = 0,1,2,  with reflecting hyperplanes T_κ,λ with equation z_1 + ζ^κ z_2 + ζ^λ z_3=0, and order 3, and

  * (ij)_κ, for 1≤ i≠ j ≤ 3 and κ=0,1,2, with reflecting hyperplanes H_i,j^κ: z_i=ζ^κz_j, and order 2. If κ=0, we may omit it from the notation.




Hyperplanes of the form H_i and T_κ,λ make up one orbit of hyperplanes which we denote by 𝒪_1, and the hyperplanes of the third type form a second orbit 𝒪_2.





  
Transversality in G_26

 Every hyperplane in 𝒪_1 is transverse with exactly 3 hyperplanes  of G_26, which all belong to 𝒪_2. Furthermore, if H,H'∈𝒪_1 are trasverse with the same 3 hyperplanes of 𝒪_2, then H=H'.



For the first statement, we show it for H_3∈𝒪_1 and the general case follows by conjugation.

First, we check the intersection H_3∩ H_1 = ⟨(0,1,0) ⟩. This is clearly a subspace of H_1,3^λ for every λ and hence H_1,H_3 are non-transverse. Also, note that H_3∩ H_1 ⊆ H_1,3^λ implies that H_3∩ H_1,3^λ ⊆ H_1 (all intersections of two hyperplanes are of co-dimension 2), and hence, H_3 is non-transverse with H_1,3^λ for all λ as well. Similarly, we obtain that H_3⋔̸H_2 and H_3⋔̸H_2,3^λ for all λ. 

We check now the intersections H_3∩ T_κ,λ =⟨ (-ζ^κ,1,0) ⟩ which can be seen to be a subspace of T_κ,λ' for every λ' ≠ λ, and so H_3, T_κ,λ are non-transverse.

Finally, the hyperplanes left to check are the hyperplanes H_1,2^λ, λ=0,1,2, which are transverse with H_3. Indeed, we have H_1,2^λ ∩ H_3 =⟨ (ζ^λ,1,0) ⟩ and one can verify that this is not contained in any hyperplane other that H_3, H_1,2^λ. So, H_3 is transverse only with H_1,2,H_1,2^1,H_1,2^2 ∈𝒪_2.

For the second statement, we verify that H_3 is the only hyperplane in 𝒪_1 transverse with H_1,2^λ,λ=0,1,2, and the result follows again by conjugation. To quickly check that, observe that the reflections t_1,t_2 permute the hyperplanes H_1,2^λ among themselves and hence, by Lemma <ref>, a hyperplane transverse with H_1,2^λ,λ=0,1,2 would stay invariant under t_1,t_2. Such a hyperplane is only H_3. 





 The orbit 𝒪_2 does not contain any pair of transverse hyperplanes.


We take again a representative H_1,2∈𝒪_2, and we check that it is not transverse with any hyperplane in 𝒪_2. For H_1,2^λ, λ=1,2, we have H_1,2∩ H_1,2^λ =⟨ (0,0,1) ⟩⊆ H_1 and so H_1,2,H_1,2^λ are non-transverse. For H_1,3^λ and any λ we have H_1,2∩ H_1,3^λ=⟨ (ζ^λ,ζ^λ,1) ⟩⊆ H_2,3^λ which yields that H_1,2⋔̸H_1,3^λ and T_1,2⋔̸T_2,3^λ as well, completing the result.


The two above lemmas imply that a transverse collection B contains at most one hyperplane from each orbit and, consequently, there are no transverse collections of cardinality more than two. As for collections of cardinality 2, since every hyperplane in 𝒪_1 is transverse with exactly 3 hyperplanes in 𝒪_2, there are 36 of them, each containing one hyperplane of 𝒪_1 and one of 𝒪_2. In fact, they all form one orbit under W, denoted ℬ. For that, notice that it is sufficient to show that all collections containing H_3 are conjugate, since every collection is conjugate to one containing H_3. Again, to see that, check that the transverse collections  {H_3,H_1,2},{H_3,H_1,2^1},{H_3,H_1,2^2} (see the proof of Lemma <ref> for a verification of their transversality) are conjugate, since t_2{H_3,H_1,2}={H_3,H_1,2^1} and t_2^2{H_3,H_1,2}={H_3,H_1,2^2}. 




  
The elements e_B for transverse collections of cardinality 2


For a transverse collection B, let (e_B) denote the annihilator of e_B in (W).

 Let B,B' be two transverse collections of G_26 of cardinality 2. If B,B' contain the same hyperplane of 𝒪_1, then (e_B)=(e_B'). 



We consider the following collections which serve as convenient representatives for what we demonstrate: B_0={H_3,H_1,2^1}, B_1={T_0,0,H_1,2},B_2={T_0,0,H_1,3},B_3={T_0,0,H_2,3}. Here one may want to verify that B_1,B_2,B_3 are indeed transverse collections. For B_1 we have T_0,0∩ H_1,2=⟨ (1,1,-2) ⟩ and one can check that this is not contained in any hyperplane other than T_0,0 and H_1,2. For B_2,B_3 the same can be obtained by conjugation from B_1 with reflections (23),(13) which stabilize T_0,0 and map H_1,2 to H_1,3 and H_1,2 to H_2,3 respectively. 

We show now that (e_B_1)=(e_B_2)=(e_B_3). For that, we compute e_T_0,0e_B_0 in two ways. First, we find the reflections taking H_3 to T_0,0; let s be such a reflection. By Lemma <ref>, the hyperplane H_s contains H_3∩ T_0,0=⟨ (1,-1,0)⟩. The only hyperplanes different from H_3, T_0,0 containing this last subspace are T_0,λ, λ=1,2 and so, the possible reflections are t_0,λ^p, λ,p∈{1,2}. To calculate which of these reflections take H_3 to T_0,0 we use the following formula for a reflection, which uses a root of the reflection (a vector perpendicular to its reflecting hyperplane). If r is a unitary reflection of V, and u∈ H_r\{0} is such that ru=αu where α is some root of unity, then for every v∈ V, r(v)=v-(1-α)⟨ v,u ⟩/⟨ u,u ⟩u.

A vector perpendicular to T_0,λ is u_λ=(1,1,ζ^λ) and since t_0,λ is the distinguished reflection of T_0,λ of order 3, then t_0,λ^p(v)=v-(1-ζ^p)⟨ v,u_λ⟩/3u_λ. Now, to check whether t_0,λ^p(H_3)=T_0,0, it suffices to check that t_0,λ^p(H_3^⊥)=T_0,0^⊥. For that, we use the vectors e_3=(0,0,1) and u_0=(1,1,1) that generate H_3^⊥ and T_0,0^⊥, respectively. Let x=-(1-ζ^p)⟨ e_3,u_λ ⟩/3=(ζ^p-1)ζ^-λ/3. We have, t_0,λ^p(e_3)=e_3+xu_λ=(0,0,1)+x(1,1,ζ^l)=(x,x,1+xζ^λ). So, t_0,λ^p(H_3)=T_0,0 if only if (x,x,1+xζ^λ)∈⟨ (1,1,1) ⟩⇔ x=1+xζ^λ ⇔ (1-ζ^λ)x=1. The last equation becomes (1-ζ^λ)(ζ_p-1)ζ^-λ=3 ⇔ ζ^p-λ+ζ^p +ζ^-λ=2. One can verify that this is true only for p=λ≠ 0, which means that there are two reflections taking H_3 to T_0,0, namely t_0,1, t_0,2^2. Let μ_1,μ_2 be the respective Brauer parameters of t_0,1, t_0,2^2. We have, 
e_T_0,0e_B_0=∑_s∈ℛ_H_3→ T_0,0 μ_sse_B_0=(μ_1t_0,1+μ_2t_0,2^2)e_B_0. Also, T_0,0 is non-transverse with H_1,2^1∈ B_0 and so e_T_0,0e_B_0=∑_s∈ℛ_H_1,2^1→ T_0,0 μ_sse_B_0=0, since H_1,2^1 and T_0,0 belong to different orbits. Combining the two, we get (μ_1t_0,1+μ_2t_0,2^2)e_B_0=0.

We write the last equation as e_t_0,1B_0μ_1t_0,1=e_t_0,2^2B_0(-μ_2t_0,2^2). Since μ_1t_0,1 and -μ_2t_0,2^2 are invertible elements of ℤ[μ^± 1]W, this implies that: 


    (e_t_0,1B_0)=(e_t_0,2^2B_0),


We find the collections t_0,1B_0, t_0,2^2B_0. Since t_0,1H_3=t_0,2^2H_3=T_0,0, we have t_0,1B_0, t_0,2^2B_0 ∈{B_1, B_2, B_3}. However, note that since t_0,1, t_0,2^2 map H_3 to T_0,0 and H_1,2 is transverse with both H_3 and T_0,0, then by Lemma <ref>, H_1,2 is invariant under t_0,1, t_0,2^2. Thus, since H_1,2∉B_0,  t_0,1B_0, t_0,2^2B_0 do not contain H_1,2 either, meaning that t_0,1B_0, t_0,2^2B_0∈{B_2,B_3} or equivalently, t_0,1H_1,2^1, t_0,2^2H_1,2^1∈{H_1,3,H_2,3}. To find hyperplanes t_0,1H_1,2^1, t_0,2^2H_1,2^1 we can either use the aforementioned formula for a unitary reflection or search in the following way. Since for any reflection s mapping H to H', we have H∩ H' ⊆ H_s (Lemma <ref>), which implies H_s ∩ H ⊆ H', in order to find the possible images of t_0,1H_1,2^1, we can first check which hyperplanes among H_1,2,H_1,3,H_2,3 contain H_1,2^1∩ T_0,1=⟨ (ζ,1,-1-ζ^-1)⟩= ⟨ (ζ,1,ζ)⟩. We see that it is only H_1,3 and so t_0,1H_1,2^1=H_1,3. In the same way, the intersection H_1,2^1∩ T_0,2=⟨ (ζ,1,-ζ^-1-ζ^-2)⟩= ⟨ (ζ,1,1)⟩ is contained only in H_2,3 and hence, t_0,2^2H_1,2^1=H_2,3. So, t_0,1B_0=B_2, t_0,2^2B_0=B_3 and Equation (<ref>) gives (e_B_2)=(e_B_3). Now, observe that for the reflection (23), which corresponds to permutation of the coordinates z_2,z_3, we have (23)B_2=B_1 and (23)B_3=B_3 and hence, conjugating Equation (<ref>) by (23) we get (e_B_1)=(e_B_3). Thus, we finally obtain (e_B_1)=(e_B_2)=(e_B_3). 

Now, since B_1,B_2,B_3 are all the collections of cardinality 2 containing T_0,0∈𝒪_1 (Lemma <ref>), by conjugation, we obtain the result of the statement.





 Let B be a transverse collection of G_26 of cardinality 2. Then e_B=0.



We first consider a product e_b'e_{b,a} where b', b∈𝒪_1 and a∈𝒪_2 is transverse with b. Of course, if b'∈ B then e_b'e_{b,a}=δe_{b,a}; otherwise, recall that every hyperplane in 𝒪_1 is determined by the hyperplanes in 𝒪_2 with which it is transverse (Lemma <ref>), and so if b≠ b', then there exists a'∈𝒪_2 transverse with b but not with b'. This gives that e_b'e_{b,a'}=∑_s∈ℛ_a'→ b'μ_sse_{b,a'}=0 since b' and a' belong to different orbits. But (e_{b,a})=(e_{b,a'}) and so e_b'e_{b,a}=0. Summing up, for all b,b'∈𝒪_1, and a∈𝒪_2 transverse with b, we have:

    e_b'e_{b,a}=δ_b',b· δe_B.

Now, we compute e_H_1,3e_B_0, where B_0={H_3,H_1,2^1} in two ways, as before, to obtain an invertible element that annihilates e_B' for some conjugate B' of B. For a verification of the mentioned transversality relations one may go back to the proof of Lemma <ref>. First of all, H_1,3⋔̸H_3, and since H_1,3,H_3 belong to different orbits, e_H_1,3e_B_0=0. In addition to that, H_1,3 is non-transverse with H_1,2^1 and so e_H_1,3e_B_0=∑_s∈ℛ_H_1,2^1→ H_1,3 μ_ss e_B_0, which yields that ∑_s∈ℛ_H_1,2^1→ H_1,3 μ_ss e_B_0=0. We rewrite this as 


    ∑_s∈ℛ_H_1,2^1→ H_1,3 e_sB_0μ_ss =0.





One can quickly see that (32)_1 is a reflection mapping H_1,2^1 to H_1,3, and H_3 to H_2. By Lemma <ref>, we know that there is at most one such reflection, so for each s mapping H_1,2^1 to H_1,3 and different from (32)_1, we have sH_3≠ H_2. This means that for such s, the hyperplane H_2 is not contained in sB_0 and so e_H_2e_sB_0=0 (Equation (<ref>)). So, multiplying Equation (<ref>) by e_H_2 on the left, we obtain
∑_s∈ℛ_H_1,2^1→ H_1,3e_H_2 e_sB_0μ_ss =0, which yields δe_(32)_1B_0μ_(32)_1(32)_1=0, or, equivalently,
e_(32)_1B_0 =0. 

Since, as we showed earlier, all transverse collections of cardinality 2 belong to the same orbit, this implies that e_B=0.





  
Conclusion
Finally, we can prove the following result.


 Let W=G_26. The algebra (W) is a free 𝒜-module, and the set {we_H| H∈ℋ, w∈ W/W_H} is a basis.

 By Theorem <ref>, for any proper field K, the set {we_B | B∈𝒞^K_adm, w∈ W/K_B} is a basis for ^K(G_26). As we showed in the discussion preceding the proposition, in G_26 there are transverse collections of cardinality at most 2 and e_B=0 for all transverse collections of cardinality 2. Thus, the above basis yields, in fact, the basis {we_H | H∈ℋ, w∈ W/K_{H}}. By Remark <ref>, for every H∈ℋ, the group K_{H} is just the pointwise stabilizer W_H of H. Hence, the set {we_H| H∈ℋ, w∈ W/W_H} is a basis of ^K(W). We show that it also spans (W), which then implies the result.


By Lemma <ref>, the set {we_B | B∈𝒞, w∈ W } spans (W). Again, by the discussion before the proposition, there are no transverse collections of cardinality more than 2, and e_B=0 if B has cardinality 2. Hence, the set {we_H| H∈ℋ, w∈ W} spans (W). Now, for every H∈ℋ and w_1,w_2 in the same left coset of W_H, i.e. w_2^-1w_1 ∈ W_H, by (B3) of Definition <ref> of the Brauer-Chen algebra, we have that w_2^-1w_1e_H=e_H, or, equivalently, w_1e_H=w_2e_H. This implies that set {we_H| H∈ℋ, w∈ W/W_H} spans (W), which concludes the proof.






















§ APPENDIX









 §.§ Calculation of the order of K_B for the infinite series


We calculate here the order of the group K_B that completes the proofs of Propositions <ref> and <ref>. Recall that, in the setting of G(m,p,n), ζ denotes the primitive m-th root of unity e^2πi/n and z_1,… ,z_n the standard coordinates of ℂ^n. We begin with a useful remark.


A matrix of G(m,p,n) can be written uniquely as the product of a permutation matrix and a diagonal matrix. In fact, if D(m,p,n) denotes the set of all diagonal ones, then we have G(m,p,n)=G(1,1,n)⋉ D(m,p,n), where G(1,1,n) coincides with the subgroup of permutation matrices. For a matrix A∈ G(m,p,n) let P_A,D_A denote the unique permutation and diagonal, respectively, matrices for which A=P_AD_A.





















Let W=G(m,p,n) and B be transverse collection of the form {H_i_1j_1^κ_1,… ,H_i_rj_r^κ_r}. Then K_B consists of all matrices in (B) whose product of non-zero entries is 1 and which leave invariant the coordinates corresponding to the indices not appearing in any hyperplane in B. Furthermore, the order of K_B is 2^rm^r-1 r!.





 First, consider B of the form {H_i_1j_1,… ,H_i_rj_r}. In the proof of Proposition <ref> we identified K_B with the group K_B^0  generated by {(ij)| H_ij∈ B}∪{(ii')_κ(j'j)_κ| H_ij',H_i'j∈ B,κ∈ℤ}, which we write more concisely as

    K_B^0=⟨ (ij),(ii')_κ(jj')_κ| H_ij,H_i'j'∈ B, κ ∈ℤ⟩.

Notice that the element (ii')_κ (jj')_κ can be written as:


    (ii')(jj')·(1,…,(i)ζ^κ, 1, … ,(j)ζ^κ,1, … ,(i')ζ^-κ,1, … ,(j')ζ^-κ,1,…).

Denote this element by h_κ(i,i',j',j), and by m_κ(i,i',j',j) the above diagonal matrix so that
h_κ(i,i',j,j')=(ii')(jj')m_κ(i,i',j,j'). Then h_0(i,i',j,j')=(ii')(jj'), and hence, h_κ(i,i',j,j')=h_0(i,i',j,j')m_κ(i,i',j,j'). So, we have that


    K^0_B=⟨(ij),(ii')(jj'),m_κ(i,i',j,j')| H_ij,H_i'j'∈ B⟩.


Now, consider the subgroups 

    K_B^(1)=⟨(ij),(ii')(jj')| H_ij,H_i'j'∈ B⟩ and  K_B^(2)=⟨ m_κ(i,i',j,j')| H_ij,H_i'j'∈ B⟩,

which generate K_B^0. One can see that K_B^(1) consists of all permutation matrices that stabilize B and  which induce the identity permutation on the coordinates z_i for the indices i that do not appear in any hyperplane in B. The order of this group is, hence, equal to the number of permutations of the indices appearing in B that leave B stable, which is 2^rr!.

For the group K_B^(2), a quick verification yields that it consists of all diagonal matrcices (ζ^λ_1,… ,ζ^λ_n) for which λ_i=λ_j for all H_ij∈ B, λ_i=0 for all i not appearing in B, and such that ∑_i=1^n λ_i=0. It follows that its order is m^r-1. 

Now, K_B^(1), K_B^(2) consist, respectively, of permutation and diagonal matrices and generate K_B^0. By Remark <ref> above, this implies that K^0_B=K_B^(1)⋉ K_B^(2), and hence, the order of K_B^0 is 2^rm^r-1r!.

For the description of K_B^0 as a subgroup of (B), one needs only verify that a matrix A=P_AD_A∈ G(m,p,n), with D_A=(ζ^λ_1,… ,ζ^λ_n), stabilizes B if and only the permutation induced by P_A on the indices appearing in B stabilizes B and λ_i=λ_j for all H_ij∈ B. Comparing this with the description of K_B^(1), K_B^(2) above, gives the characterization of the statement for B.

Finally, as in the proof of Proposition <ref>, a general transverse collection of the form {H_i_1j_1^κ_1,… ,H_i_rj_r^κ_r} is the image of B under the element ∏_i=1^rt̂_i^κ_i∈ G(m,1,n). Furthermore, as explained in the same proof, K_gB=gK_Bg^-1 for all g∈ G(m,1,n). This yields the order of K_B for all such transverse collections. 

Moreover, the product of the non-zero entries of a monomial matrix A=P_AD_A∈ G(m,p,n) is equal to its determinant times the sign of the permutation corresponding to P_A matrix, and is, hence, invariant under conjugation by G(m,1,n). So A∈(B) has product of non-zero entries equal to 1 if and only if the same is true for gAg^-1∈ g(B)g^-1=(gB). Similarly, A leaves invariant the coordinates corresponding to the indices not appearing in B if and only if the corresponding property is true for the matrix gAg^-1 and the coordinates corresponding to the indices not appearing in the transverse collection gB. This implies that the characterization of the statement is invariant under conjugation by elements of G(m,1,n), and is, hence, true for a general collection of the form {H_i_1j_1^κ_1,… ,H_i_rj_r^κ_r}. This concludes the proof.

















Let W=G(2,2,n) and B={H_i_1j_1, H_i_1j_1^1,… ,H_i_rj_r, H_i_rj_r^1} (with all indices distinct). Then K_B consists of all matrices in (B) that are diagonal on the lines and rows corresponding to the indices not appearing in any hyperplane in B. Furthermore, its order is 2^r+n-1r!.



 

In the proof of Proposition <ref> we identified K_B with the following subgroup of G(m,p,n):


    K_B^0=⟨ (ij_1)_κ, (i_1i)_κ(i_1i)_κ+1,(i_1i)_κ(j_1j)_κ, (j_1j_2)_κ(j_1j_2)_κ+1⟩,

with H_ij_1, H_i_1j∈ B, κ∈ℤ, and  1≤ j_2 ≤ n. Notice that elements of the second form in the above presentation are obtained by all elements of the fourth form. This, together with a change of variables for a clearer exposition, yields the following shorter presentation of K_B^0:


    ⟨ (ij)_κ, (ii_1)_κ(jj_1)_κ, (j_1j_2)_κ(j_1j_2)_κ+1| H_ij,H_i_1j_1∈ B, κ∈ℤ, 1≤ j_2 ≤ n⟩



As in the proof of Proposition <ref> above, we have


    (ii_1)_κ(jj_1)_κ=(ii_1)(jj_1)·(1,…,(i)ζ^κ, 1, … ,(j)ζ^κ,1, … ,(i_1)ζ^-κ,1, … ,(j_1)ζ^-κ,1,…),

and we set h_κ(i,i_1,j_1,j)=(ii_1)_κ(jj_1)_κ and m_κ(i,i_1,j,j_1) the above diagonal matrix, so that h_κ(i,i_1,j_1,j)
=(ii_1)(jj_1)· m_κ(i,i_1,j,j_1). Note, again that (ii_1)(jj_1)=h_0(i,i_1,j_1,j). One can also verify the following two identities,


    (ij)_κ=(ij)·(1,…,(i)ζ^κ, 1, … ,(j)ζ^κ,1, … ,1),

and 

    (j_1j_2)_κ(j_1j_2)_κ+1=(1,…,(j_1)ζ, 1, … ,(j_2)ζ^-1,1, … ,1).
 
Let l_κ(i,j)=(1,…,(i)ζ^κ, 1, … ,(j)ζ^κ,1, …) and q_κ(j_1,j_2)=(j_1j_2)_κ(j_1j_2)_κ+1, and consider the following subgroups of K_B^0: 


    K_B^(1)=⟨ (ij),(ii_1)(jj_1)| H_ij,H_i_1j_1∈ B ⟩

and

    K_B^(2)=⟨ l_κ(i,j), q_κ(j_1,j_2), m_κ(i,i_1,j,j_1) | H_ij,H_i_1j_1∈ B, κ∈ℤ, 1≤ j_2 ≤ n  ⟩.

By the above, it is clear that the subgroups K_B^(1), K_B^(2) generate K_B. Furthermore, in view of Remark <ref>, since they consist, respectively, of permutation and diagonal matrices, then K_B=K_B^(1)⋉ K_B^(2).


Now, one can see that K_B^(1) consists of all permutation matrices stabilizing B and inducing the identity permutation on the coordinates z_i for i not appearing in any hyperplane in B. Thus, its order equals the number of permutations of the indices appearing in B that stabilize B, which is 2^rr!.

For K_B^(2), one can, in fact, verify that it consists of all diagonal matrices (ζ^λ_1, … ,ζ^λ_n) for which ∑_i=1^nλ_i=0, i.e. all diagonal matrices in G(m,p,n); this group has order 2^n-1. This yields the order of K_B^0, which is 
2^n+r-1r!.

For the characterization of K_B^0 as a subgroup of (B), since all diagonal matrices stabilize B, we notice that a matrix A=P_AD_A∈ G(2,2,n) stabilizes B if and only if the permutation matrix P_A stabilizes B. By the above, the group K_B^0 contains, as (B), all diagonal matrices of G(m,p,n) together with all permutation matrices stabilizing B, with the extra condition that they induce the identity permutation on the coordinates z_i for indices i not appearing in any hyperplane in B. One can see that this yields the characterization of the statement. 






 §.§ Table of admissibility and dimensions for the exceptional groups



The following table contains the information of admissibility for all exceptional complex reflection groups. Each row corresponds to an orbit of transverse collections. The first column shows the cardinality of a representative of the orbit and the second the cardinality of the orbit. In the third column, we mention the size of the quotient (B)/K_B.  If the corresponding orbit consists of non-admissible collections, then we mark 0 in this column. If for some cardinality there are no admissible collections of that cardinality, we do not include the corresponding row in the table at all. Finally, in the last column we mention the dimension of the Brauer-Chen algebra, as given by the formula:



    | W | + ∑_B|ℬ|^2 ·| S_B / K_B |,
 
where B runs through a set of representatives from each orbit of transverse collections (this formula is directly implied by the formula of Theorem <ref>). For the groups G_25 and  G_32, we include with an asterisque the corresponding data for proper fields where μ∈𝕌_6 (recall that μ is the quotient of the parameters corresponding to the two orbits of reflections in these groups). 













9
















BMW
J. Birman, H.Wenzl, 
Braids, Link Polynomials and a New Algebra.
Transactions of the American Mathematical Society 313, (1989), 249-273

BoC
C. Bowman, A. G. Cox,
Decomposition numbers for Brauer algebras of
type G(m, p, n) in characteristic zero.
Journal of Pure and Applied Algebra 218, (2014), 992-1002.












BMR
M. Broué, G. Malle, R. Rouquier
Complex reflection groups, braid groups, Hecke algebras.
Journal für die reine und angewandte Mathematik 500 (1998), 127-190



Br
R. Brauer,
On algebras which are connected with semisimple continuous groups. 
Annals of Mathematics, Second Series, 38, (1937), 857-872



Ch
Z. Chen,
Flat connections and Brauer type algebras.
Journal of Algebra 365, (2012), 114-146

ChBMW
Z. Chen,
Birman-Murakami-Wenzl type algebras for arbitrary Coxeter systems.
preprint 2017, arXiv:1708.04833


CFW 
A.M. Cohen, B. Frenk, D.B. Wales,
Brauer algebras of simply laced type.
Israel Journal of Mathematics 173, (2009), 335-365


CGW
A.M. Cohen, D.A.H. Gijsbers, D.B. Wales,
BMW algebras of simply laced type.
Journal of Algebra 286, (2005), 107-153

CL
A.M. Cohen, S. Liu
Brauer algebras of type B.
Forum Mathematicum 27 (2015), 1163-1202


CLY
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