The Q-Brauer Algebras

The q-Brauer algebras

Von der FakultätMathematik und Physik der UniversitätStuttgart zur Erlangung der Würdeeines Doktors der Naturwissenschaften (Dr. rer. nat) genehmigte Abhandlung

Vorgelegt von Nguyen Tien Dung

aus Vinh, Vietnam

Hauptberichter: Prof. Dr. rer. nat. S. K¨onig Mitberichter: Prof. Dr. rer. nat. R. Dipper

Tag der m¨undlichen Pr¨ufung: 23. April 2013

Institut f¨urAlgebra und Zahlentheorie der Universit¨atStuttgart

2013 D93 Diss. Universit¨atStuttgart Eidesstattliche Erkl¨arung

Ich versichere, dass ich diese Dissertation selbst¨andigverfasst und nur die angegebenen Quellen und Hilfsmittel verwendet habe.

Nguyen Tien Dung Acknowledgements

In my humble acknowledgements, it is a pleasure to express my gratitude to all people who have supported, encouraged and helped me during the time I spent working on this thesis. First of all, I would like to record my deep and sincere gratitude to my supervisor Prof. Dr. Steffen Koenig. I am grateful to him for super- vision, advice, and guidance from the early stage of this research as well as giving me valued suggestions through out the work. He provided me an open and friendly environment in the discussions and supported me in various ways that include two and a half year financial assistance and many recommendation letters, in particular, for DAAD. Furthermore, many thanks to all of my colleagues who helped me during this project. In particular, thank to Qunhua Liu for support and help at the basic steps in my work, to Inga Benner, Frederick Marks and Mrs Elke Gangl for their help dealing with administrative and language (German) matters, to Shalile Armin for the useful discussions concerning the Brauer algebra, and Qiong Guo for her help concerning LaTeX problems. For financial support I would like to thank the Project MOET - 322 of the Vietnam government and (partly) the DFG Priority Program SPP- 1489. Finally, I specially express my appreciation to my parents for their encouragement and invaluable support over the last years, to my wife and my lovely daughter for their inseparable and spiritual support.

Stuttgart, 2013 Nguyen Tien Dung Zusammenfassung

Diese Dissertation untersucht strukturelle Eigenschaften von q-Brauer Algebren über einem kommutativen Ring mit Einselement oder einem Körper der Charakteristik p ≥ 0. Wir konstruieren zunächst eine Zell-asis fürdie q-Brauer Algebra über einem kommutativen Ring mit Einselement und zeigen dann, dass die q-Brauer Algebra zellulär ist im Sinne von Graham und Lehrer. Anschließend klassifizieren wir alle einfachen Moduln bis auf Isomorphie fürden Fall der q-Brauer Algebra über einem Körper beliebiger Charakteristik. Des Weiteren geben wir einen alternativen Beweis dieser Klassifikation in kombinatorischer Sprache an. Dieser Beweis verwendet die Konstruktion einer weiteren neuen Basis der q-Brauer Algebra, welche eine Hochhebung der soge- nannten Murphy Basis der Hecke Algebra der symmetrischen Gruppe ist. Diese neue Basis erlaubt es uns auch zu zeigen, dass die q-Brauer Algebra und die BMW-Algebra als Algebren im Allgemeinen nicht isomorph sein können. Im Falle der q-Brauer Algebra über einem Körper beliebiger Charakteristik bestimmen wir weiterhin die Werte des Parameters, für die die q-Brauer Algebra quasi-erblich ist. Außer- dem zeigen wir, dass die q-Brauer Algebra zellulär stratifiziert ist durch Angabe einer entsprechenden iterierten Aufblasungsstruktur. Dies wiederum impliziert Ergebnisse über die Eindeutigkeit von Specht- Filtrierungsvielfachheiten der q - Brauer Algebra, sowie die Existenz von Young Moduln und Schur Algebren für q-Brauer Algebren. Abstract

This thesis studies structural properties of the q-Brauer algebra over a commutative ring with identity or a field of any characteristic p ≥ 0. Over a commutative ring with identity we first construct a cell basis for the q-Brauer algebra and then show that the q-Brauer algebra is a cellular algebra in the sense of Graham and Lehrer. Subsequently, we classify all simple modules, up to isomorphism, of the q-Brauer algebra over a field of any characteristic. This classification is reproved by combinatorial language after constructing another new basis for the q-Brauer algebra which is a lift of the Murphy basis of the Hecke algebra of the symmetric group. This new basis enables us to prove that in general, there does not exist an algebra isomorphism between the q-Brauer algebra and the BMW-algebra. Over a field of any characteristic we determine the choices of parameter such that the q-Brauer algebra is quasi-hereditary. Moreover, we show that the q-Brauer algebra is cellularly stratified by providing a suitable iterated inflation structure. This implies results about the uniqueness of Specht filtration multiplicities of the q-Brauer algebra, as well as the existence of Young modules and a Schur algebra of the q-Brauer algebra. Contents

Introduction iii

1 Background 1 1.1 Cellular algebras ...... 1 1.2 Cellularly stratiﬁed algebras ...... 3 1.3 Hecke algebras of the symmetric groups ...... 4 1.4 Brauer algebra ...... 10

2 The q-Brauer algebras 16 2.1 Deﬁnitions ...... 16 2.2 Basic properties ...... 20 ∗ 2.3 The Brn(r, q)-modules Vk ...... 21

3 A basis and an involution for the q-Brauer algebra 22 3.1 A basis for q-Brauer algebra ...... 22 3.2 An involution for the q-Brauer algebra ...... 24 3.3 An algorithm producing basis elements ...... 32 3.4 A comparison ...... 35

4 Cellular structure of the q-Brauer algebra 39 4.1 An iterated inﬂation ...... 39 4.2 Main results ...... 51

5 Quasi-heredity of the q-Brauer algebra over a ﬁeld 55

6 A Murphy basis 58 6.1 Main theorem ...... 59 6.2 Representation theory over a ﬁeld ...... 70 6.3 Is the q-Brauer algebra generically isomorphic with the BMW-algebra? ...... 72

7 The cellularly stratiﬁed structure of the q-Brauer algebra 77 7.1 The main result ...... 77 7.2 Consequences ...... 84

i ii CONTENTS

Bibliography 87 Introduction

The classical Schur-Weyl duality relates the representation theory of inﬁ- nite group GLN (C) with that of symmetric group Sn via the mutually cen- tralizing actions of two groups on the tensor power space (CN )⊗n. In 1937 Richard Brauer [2] introduced the algebras which are now called ’Brauer algebra’. These algebras appear in an analogous situation where GLN (C) is replaced by either a symplectic or an orthogonal group and the group algebra of the symmetric group is replaced by a Brauer algebra. There is an analogous situation of Schur-Weyl duality in quantum theory. In 1986, Jimbo [22] has shown that the actions of the quantized enveloping

algebra Uq(glN ) and the Hecke algebra Hn(q) of the symmetric group on the same space are centralizer actions. In 1989, Birman and Wenzl [1], independently Murakami [31] in 1987, have introduced an algebra related with topology. This algebra, nowadays called BMW-algebra, is a q-deformation of the Brauer algebra, and is shown to play the same role in the quantum case as the Brauer algebra in classical Schur-Weyl

duality. In detail, the quantized enveloping algebra Uq(glN ) is replaced by the quantized enveloping algebra Uq(oN ) or Uq(spN ), and the Hecke algebra Hn(q) is replaced by a BMW-algebra with suitable choice of parameters (see Leduc and Ram [27]). In the classical Schur - Weyl duality, the symplectic and orthogonal group are subgroups of the group GLN (C), and the group algebra of the symmetric group is a subalgebra of the Brauer algebra. However, in the

quantum case, both Uq(oN ) and Uq(spN ) are not known to be subalgebras of Uq(glN ). Similarly, we only know that Hn(q) is isomorphic to a quotient of the BMW-algebra. Recently, a new algebra, another q-deformation of the Brauer algebra, has been introduced via generators and relations by Wenzl [39] who called

it the q-Brauer algebra. This new algebra contains Hn(q) as a subalgebra, and over the ﬁeld Q(r, q) it is semisimple and isomorphic to the Brauer algebra. It is expected, by Wenzl, to play the same role as that of the BMW-algebra in the quantum case. In particular, the q-Brauer algebra

should correspond to a q-deformation of the subalgebra U(soN ) ⊂ U(slN ) in the quantum Schur - Weyl duality. Over the complex ﬁeld C some applications of the q-Brauer algebra were found by Wenzl in [40] and [41]. iv

In [29] Molev has introduced an algebra which has close relation with the q-Brauer algebra. He has shown that the q-Brauer algebra is a quotient of his algebra in [30]. However, there has been no detailed research about the q-Brauer algebra over an arbitrary field of any characteristic so far. Even over the field of characteristic zero structure of the q-Brauer algebra is not known. The main aim of this thesis is to investigate structural properties, as well as to give a complete classification of isomorphism classes of simple modules of the q-Brauer algebra over a field R of any characteristic. The first main result presented here is the following:

Theorem 4.2.2. Suppose that Λ is a commutative noetherian ring which contains R as a subring with the same identity. If elements q, r and

(r − 1)/(q − 1) are invertible in Λ, then the q-Brauer algebra Brn(r, q) over the ring Λ is cellular with respect to an involution i. For the proof of this theorem, we will first construct an explicit basis and provide an involution i for the q-Brauer algebra (Theorem 3.1.4 and Proposition 3.2.2, respectively). The basis constructed is indexed by diagrams of the classical Brauer algebra, and is helpful for producing cellular structure of the q-Brauer algebra. It should be noticed that Wenzl [39] has introduced a generic basis for the q-Brauer algebra. But, it seems that his basis is not suitable to provide a cell basis (a detailed discussion of this is in Chapter 3, Section 3.4). Then, using an equivalent definition with that of Graham and Lehrer, given by Koenig and Xi in [24] and their approach, called ’iterated inflation’, to cellular structure ([26] or [24]), we show that the q-Brauer algebra is an iterated inflation of Hecke algebras H2k+1 of the ∗ symmetric group along vector spaces Vk,n and Vk,n for k = 0, 1, 2,... [n/2] in Proposition 3.2.2. Applying the cellularity of the q-Brauer algebra and a result due to Dipper and James for the Hecke algebra of symmetric group ([7], Theorem 7.6), we can classify the simple modules, up to isomorphism, of the q-Brauer algebra. The second main result is the following.

Theorem 4.2.7 (reproved by combinatorics in Theorem 6.2.1)

Let Brn(r, q) be a q-Brauer algebra over an arbitrary ﬁeld R with characteristic p ≥ 0. Moreover assume that q, r and (r − 1)/(q − 1) are invertible in R. Then the non-isomorphic simple Brn(r, q)-modules are parametrized by the set {(n − 2k, λ) ∈ I| λ is an e(q)-restricted partition of n − 2k}.

In this thesis, we give two different proofs for the last result. One proof is based on an iterated inflation structure of the q-Brauer algebra. The other one, however, uses combinatorial language which does not relate to the iterated inflation structure. In detail, the second proof is done by using v an R-bilinear form on a Murphy basis of the Specht module C(k, λ). To do this, we will construct another basis which is a lift of the Murphy basis of the Hecke algebra of the symmetric group. The third main result of this thesis is as follows.

2 2 Theorem 6.1.10 The q-Brauer algebra Brn(r , q ) is freely generated as an R–module by the collection of basis elements lifted from the Murphy basis of the Hecke algebra. A basis element is indexed by two pairs, in each pair the ﬁrst entry is a standard tableaux and the second one is a certain partial Brauer diagram. Moreover, the following statements hold.

1. The involution i swaps the indices in each basis element.

2. The multiplication of basis elements yields a cellular basis.

This basis is called a Murphy basis for the q-Brauer algebra. The Murphy basis enables us to answer the negative question: Are, in general, the q-Brauer algebra and the BMW-algebra isomorphic? (see Section 6.3).

Motivated by work of Hemmer and Nakano [20] and Hartmann and Paget [18], in [19] Hartmann, Henke, Koenig and Paget have introduced the concept ”cellularly stratified algebra” for a class of diagram algebras including the Brauer algebra, the BMW-algebra, and the partition algebra. We do not know whether the q-Brauer algebra is a diagram algebra. How- ever, we will prove that the q-Brauer algebra is a cellularly stratified algebra by providing a suitable iterated inflation structure (Theorems 7.1.7, 7.1.8). Combining this result and a result due to Hemmer and Nakano [20] relat- ing the Specht filtration multiplicity of the Hecke algebra of the symmetric group, we derive some results about the Specht filtration multiplicity of the q-Brauer algebra, as well as the existence of Young modules and the q-Schur algebra of the q-Brauer algebra (Theorem 7.2.3 and 7.2.4).

In Chapter 1 we recall the background of this thesis. We first review definitions of cellular and cellularly stratified algebras. Then, in Section 1.3 we describe some combinatorics, as well as some basic facts about the Hecke algebra of the symmetric group that are used in this document, such as the Murphy basis, a criterion for being semisimple, and a condition that the Specht filtration multiplicity is well-defined. Section 1.4 focuses on the Brauer algebra and its fundamentals. In particular, we will recall Wenzl’s definition of the length function for the Brauer algebra. This definition and Lemmas 1.4.8, 1.4.9 are crucial for constructing an explicit basis of the q-Brauer algebra in Chapter 3.

Chapter 2 starts with recalling the original definitions of the q-Brauer algebra. Then, we give slightly more general and more flexible modified vi

definitions (Definitions 2.1.4-2.1.7) that will be used in this thesis. Some discussions which indicate relations between the q-Brauer algebra and Molev’s algebra (resp. the Brauer algebra), as well as between the versions of the q-Brauer will be also mentioned in this section. Sections 2.2 and 2.3 collect some properties of the q-Brauer algebra that are needed for later references. Starting from Chapter 3, all results stated are new. The first section in Chapter 3 contains a construction of an explicit basis for the q-Brauer algebra. In Section 3.2, by extending more properties we prove that there exists an involution for the q-Brauer algebra (Proposition 3.2.2). Notice that this involution has first been defined by Wenzl in his preprint article [38] without proving. Then, our proof for this (Lemma 3.2.1) has been recognized and quoted by Wenzl in the published article [39], Lemma 3.3(g). In Section 3.3 an algorithm tells us how to write out a basis element of the q-Brauer algebra from a given particular diagram. Finally, we will give some examples in Section 3.4 to show that Wenzl’s generic basis for the q-Brauer algebra is not suitable to provide a cell basis. Chapter 4 is devoted to describing the cellular structure of the q-Brauer algebra. In Section 4.1, we will first show that there is a bijection from the q-Brauer algebra to a direct sum of tensor spaces (Lemma 4.1.7). Then, we prove that the involution of the q-Brauer algebra induces an involution on the tensor space, and this induced involution satisfies particular conditions (Lemma 4.1.14). Hence, we obtain the result in Proposition 4.1.15 that the q-Brauer has an iterated inflation. Main results of thesis are Theorems 4.2.2 and 4.2.7 in Section 4.2. In Chapter 5 we will prove a necessary and sufficient condition for the q-Brauer algebra over an arbitrary field to be quasi-hereditary (Theorem 5.0.12). This result is analogous to those for the Brauer algebra and the BMW-algebra [25]. In Chapter 6, after giving a Murphy basis for the q-Brauer algebra in Theorem 6.1.10 we apply it to solve two problems: 1. Give a combinatorial proof for Theorem 4.2.7. 2. Show that in general, there does not exist an algebra isomorphism between the q-Brauer algebra and the BMW-algebra (Claim 6.0.19). In the final chapter, the q-Brauer algebra will be proved to be cellularly stratified (Theorem 7.1.8). Then, some consequences of this result are stated in Theorems 7.2.3 and 7.2.4. Finally, we remark that all results in this thesis for the q-Brauer algebra recover those of the Brauer algebra if q = 1 or q → 1 when this makes sense. Results from Chapter 3 to Chapter 5 in this dissertation have been presented in the article [11]. Chapter 1

Background

1.1 Cellular algebras

In this section we recall the original definition of cellular algebras in the sense of Graham and Lehrer in [15] and an equivalent definition given in [23] by Koenig and Xi. Then we are going to apply the idea and approach in [26, 43] to cellular structure to prove that the q-Brauer algebra Brn(r, q) is cellular in Chapter 4 (Theorem 4.2.2. Using the result in this section we can determine when the q-Brauer algebras is quasi-hereditary in Chapter 5. Definition 1.1.1. (Graham and Lehrer [15]) Let R be a commutative Noetherian integral domain with identity. A cellular algebra over R is an associative (unital) algebra A together with cell datum (Λ, M, C, i), where (C1) Λ is a partially ordered set (poset) and for each λ ∈ Λ, M(λ) is a λ finite set such that the algebra A has an R-basis CS,T , where (S,T ) runs through all elements of M(λ) × M(λ) for all λ ∈ Λ.

(C2) Let λ ∈ Λ and S,T ∈ M(λ). Then i is an involution of A such that λ λ i(CS,T ) = CT,S. (C3) For each λ ∈ Λ and S,T ∈ M(λ) then for any element a ∈ A we have

λ X λ aCS,T ≡ ra(U, S)CU,T (mod A(< λ)), U∈M(λ)

where ra(U, S) ∈ R is independent of T, and A(< λ) is the µ 0 0 R-submodule of A generated by {C 0 0 |µ < λ; S ,T ∈ M(µ)}. S ,T

λ The basis {CS,T } of a cellular algebra A is called a cell basis. In [15], Graham and Lehrer deﬁned a bilinear form φλ for each λ ∈ Λ with respect to this basis as follfows.

λ λ λ CS,T CU,V ≡ φλ(T,U)CS,V (mod A < λ). 2 Background

When R is a ﬁeld, they also proved that the isomorphism classes of simple modules are parametrized by the set

Λ0 = {λ ∈ Λ| φλ 6= 0}.

The following is an equivalent deﬁnition of cellular algebra.

Deﬁnition 1.1.2. (Koenig and Xi [23]) Let A be an R-algebra where R is a commutative noetherian integral domain. Assume there is an involution i on A. A two-sided ideal J in A is called cell ideal if and only if i(J) = J and there exists a left ideal ∆ ⊂ J such that ∆ is ﬁnitely generated and free over R and such that there is an isomorphism of A-bimodules α :

J ' ∆ ⊗R i(∆) (where i(∆) ⊂ J is the i-image of ∆) making the following diagram commutative:

α J / ∆ ⊗R i(∆)

i x⊗y7→i(y)⊗i(x) α J / ∆ ⊗R i(∆) The algebra A with the involution i is called cellular if and only if 0 0 0 there is an R-module decomposition A = J1 ⊕ J2 ⊕ ...Jn (for some n) with 0 0 j 0 i(Jj) = Jj for each j and such that setting Jj = ⊕l=1Jl gives a chain of two-sided ideals of A: 0 = J0 ⊂ J1 ⊂ J2 ⊂ ... ⊂ Jn = A (each of them fixed 0 by i) and for each j (j = 1, ..., n) the quotient Jj = Jj/Jj−1 is a cell ideal (with respect to the involution induced by i on the quotient) of A/Jj−1. Recall that an involution i is defined as an R-linear anti-automorphism 2 0 of A with i = id. The ∆ s obtained from each section Jj/Jj−1 are called cell modules of the cellular algebra A. Note that all simple modules are obtained from cell modules [15]. In [23], Koenig and Xi proved that the two definitions of cellular algebra are equivalent. The first definition can be used to check concrete examples, the latter, however, is convenient to look at the structure of cellular algebras as well as to check cellularity of an algebra. Typical examples of cellular algebras are the following: Group algebras of symmetric groups, Hecke algebras of the symmetric group algebra or even of Ariki-Koike of finite type [14] (i.e., cyclotomic Hecke algebras), Schur algebras of type A, Brauer algebras, Temperley-Lieb and Jones algebras which are subalgebras of the Brauer algebra [15], partition algebras [42], BMW-algebras [43], and recently Hecke algebras of finite type [14]. 1.2. Cellularly stratified algebras 3

1.2 Cellularly stratiﬁed algebras

This section reviews an axiomatic definition of cellularly stratified algebras and some statements given in [19] by Hartman, Henke, Koenig, and Paget. Let A be an algebra (with identity) which can be realized as an iterated inflation of cellular algebras Bl along vector spaces Vl for k = 1, . . . , n. By [24] Section 3.1, this implies that as a vector space

n M A = Vk ⊗F Vk ⊗F Bk, (1.2.1) k=1 and A is cellular with a chain of two-sided ideals

{0} = J0 ⊆ J1 ⊆ ...Jn = A, which can be reﬁned to a cell chain, and each subquotient Jk/Jk−1 equals Vk ⊗F Vk ⊗F Bk as an algebra without unit. The involution i in A, an 2 anti-automorphism with i = id, is deﬁned through the involution jk of the cellular algebra Bk where

i(u ⊗ v ⊗ b) = v ⊗ u ⊗ jk(b) (1.2.2) for any b ∈ Bk and u, v ∈ Vk. The multiplication rule of a layer Vk ⊗F Vk ⊗F Bk is dictated by the axioms of inﬂation and given by

(x ⊗ y ⊗ b) · (x0 ⊗ y0 ⊗ b0) = (x ⊗ y0 ⊗ bϕ(y, x0)b0) + lower terms, (1.2.3)

0 0 0 for x, x , y, y ∈ Vk and b, b ∈ Bk, where ϕ is the bilinear form coming with the inflation data. Here lower terms refers to the elements in the lower layers Vl ⊗F Vl ⊗F Bl for l < k. Let 1Bk be the unit element of the algebra Bk. We define: Definition 1.2.1. A finite dimensional associative algebra A over a field

R is called cellularly stratified with stratification data (V1,B1,...,Bn,Vn) if and only if the following conditions are stratified:

(E) For each k = 1, . . . , n there exist non-zero elements uk, vk ∈ Vk such that

e(k) = uk ⊗ vk ⊗ 1Bk is an idempotent. 4 Background

(I) If l > k, then e(k)e(l) = e(k) = e(l)e(k).

Lemma 1.2.2. Let A be cellularly stratiﬁed and 1 ≤ k ≤ n. The following holds:

1. The ideal Jk is generated by e(k), that is , Jk = Ae(k)A.

2. The algebra A/Jk is cellularly stratiﬁed.

Lemma 1.2.3. Let A be cellularly stratiﬁed. With the set-up as in Deﬁ- nition 1.2.1, there is an algebra isomorphism Bk ' e(k)Ae(k)/e(k)Jk−1e(k) with 1Bk mapped to e(k).

In [19] Hartman, Henke, Koenig and Paget proved that the classical Brauer algebra, BMW-algebra, and partition algebra are cellularly stratified algebras. All these algebras are ’diagram algebras’, meaning that they have a basis which can be represented by certain diagrams and a multiplication rule using modified concatenation. In particular, let R be an arbitrary field of any characteristic p ≥ 0. Then, the statements are the following. Statement 1. Let R be an field, n an integer and N ∈ R. If r is even, −1 suppose that N 6= 0. Then the BMW-algebra BMWF (n, λ, q − q ,N) is cellularly stratified. Statement 2. Let R be an field, n an integer and N ∈ R. Suppose that N 6= 0. Then the partition algebra PF (n, N) is cellularly stratified. Statement 3. Let R be an field, n an integer and N ∈ R. If n is even, suppose that N 6= 0. Then the Brauer algebra Dn(N) is cellularly stratified. For definitions of BMW-algebra and Partition algebra we refer the reader to [1] and [42] for detail. The Brauer algebra will be introduced in Section 1.4.

1.3 Hecke algebras of the symmetric groups

This section reviews combinatorics of tableaux and some basic facts on the representation theory of the Hecke algebra of the symmetric group. Details can be found in the papers of Dipper and James [7], Murphy [31], or the book by Mathas [28]. Combinatorics and Tableaux

Throughout, n will denote positive integer, and Sn will be the symmetric group acting on {1, . . . , n} on the right. For i an integer, 1 ≤ i < n, let si 1.3. Hecke algebras of the symmetric groups 5

denote the transposition (i, i + 1). Then Sn is generated by s1, s2, . . . , sn−1, which satisfy the deﬁning relations

2 si = 1 for 1 ≤ i < n;

sisi+1si = si+1sisi+1 for 1 ≤ i < n − 1;

sisj = sjsi for 2 ≤ |i − j|.

Let k be an integer, 0 ≤ k ≤ [n/2]. Denote S2k+1,n to be the subgroup of Sn generated by generators s2k+1, s2k+2, ··· , sn−1. For i, j intergers, 1 ≤ i, j ≤ n, denote si,j := sisi+1 . . . sj if i ≤ j and si,j := sisi−1 . . . sj if otherwise.

Deﬁnition 1.3.1. Let w be a permutation in Sn. An expression

w = si1 si2 ··· sim in which m is minimal is called a reduced expression for w, and `(w) = m is the length of w. Deﬁnition 1.3.2. (1) Let e(q) be the least positive integer m such that 2 m−1 [m]q = 1 + q + q + ... + q = 0 if that exists, and let e(q) = ∞ otherwise. (2) Similarly, let e(q2) be the least positive integer m such that 2 4 2(m−1) 2 [m]q2 = 1 + q + q + ... + q = 0 if it exists, and let e(q ) = ∞ otherwise. Deﬁnition 1.3.3. Let k be an integer, 0 ≤ k ≤ [n/2]. If n − 2k > 0, a

partition of n − 2k is a sequence λ = (λ1, λ2, ··· ) of non-negative integers P such that λi ≥ λi+1 for all i ≥ 1 and |λ| = i=1 λi = n − 2k. The integers λi, for i ≥ 1, are the parts of λ; if λi = 0 for i > m we identify λ with (λ1, λ2, ··· , λm) and denote λ ` n − 2k. If n − 2k = 0, write λ = ∅ for the empty partition.

Definition 1.3.4. A partition λ = (λ1, λ2, ..., λf ) of n − 2k is called 2 e(q) − restricted if λi − λi+1 < e(q) for all i ≥ 1. The e(q ) − restricted partition is defined similarly. Definition 1.3.5. For k, n non-negative integers, 0 ≤ k ≤ [n/2], let λ be a partition of n − 2k. The Young diagram of a partition λ is the subset

[λ] = {(i, j): λi ≥ j ≥ 1 and i ≥ 1 } ⊆ N × N\{0, 0}. The elements of [λ] are the nodes of λ and more generally a node is a pair (i, j) ∈ N × N\{0, 0}. The diagram [λ] is represented as an array of boxes with λi boxes on the i–th row. For example, if λ = (3, 1), then [λ] = . If the coordinate of the box p is (i, j), deﬁne the content of p by c(p) = j − i. We say a partition µ is contained in the partition λ and write µ ⊂ λ if µi ≤ λi for all i. Deﬁne Y (λ/µ) to be the sub-diagram of Y (λ), which consists of the boxes in Y (λ) \ Y (µ). 6 Background

Deﬁnition 1.3.6. (1) Let k be an integer, 0 ≤ k ≤ [n/2], and λ be a partition of n − 2k.A λ–tableau labeled by {2k + 1, 2k + 2, . . . , n} is a bijection t from the nodes of the diagram [λ] to the integers {2k + 1, 2k + 2, . . . , n}. A given λ–tableau t :[λ] → {2k + 1, 2k + 2, . . . , n} can be visualized by labeling the nodes of the diagram [λ] with the integers 2k +1, 2k +2, . . . , n. For instance, if n = 10, k = 2 and λ = (3, 2, 1),

5 7 8 t = 6 10 (1.3.1) 9

represents a λ–tableau. (2) A λ–tableau t labeled by {2k+1, 2k+2, . . . , n} is said to be standard if the entries in t increase from left to right in each row and from top to bottom in each column. (3) Let tλ denote the λ-tableau in which the integers 2k+1, 2k+2, . . . , n are entered in increasing order from left to right along the rows of [λ]. For instance, let n = 10, k = 2 and λ = (3, 2, 1),

λ 5 6 7 t = 8 9 . 10

The tableau tλ is referred to as the superstandard tableau. (4) Deﬁne Std(λ) to be the set of all standard λ–tableaux labeled by the integers {2k + 1, 2k + 2, . . . , n}.

Deﬁnition 1.3.7. For n, k, l integers, 0 ≤ k, l ≤ [n/2], let λ and µ be partitions of n − 2k and n − 2l respectively. The dominance order on partitions is deﬁned as follows: λ ¤ µ if either

1. |µ| > |λ| or

Pm Pm 2. |µ| = |λ| and i=1 λi ≥ i=1 µi for all m > 0.

We will write λ £ µ to mean that λ ¤ µ and λ 6= µ.

For example: Let λ = (4, 1, 1) ` 6, µ = (3, 3) ` 6, ν = (3, 1) ` 4. Then ν £ λ and ν £ µ, but λ 6 ¤µ and µ 6 ¤λ. The last example implies that the dominance order ¤ is an partial order on partitions.

The symmetric group S2k+1,n acts on the set of λ–tableaux on the right in the usual manner, by permuting the integer labels of the nodes of [λ]. For example,

5 6 7 5 7 8 8 9 (6, 8, 7)(9, 10) = 6 10 . (1.3.2) 10 9 1.3. Hecke algebras of the symmetric groups 7

Definition 1.3.8. Let λ be a partition of n − 2k. λ (1) Define Young subgroup Sλ to be the row stabilizer of t in S2k+1,n. (2) For t any λ–tableau, define d(t) to be a permutation in Sλ such that

t = tλd(t).

For instance, when n = 10, k = 2 and λ = (3, 2, 1), then a direct calculation yields Sλ = hs5, s6, s8i. Using the tableau t in (1.3.1) above it deduces that d(t) = (6, 8, 7)(9, 10) by (1.3.2).

The Hecke algebra

Definition 1.3.9. Let R be a commutative ring with identity 1, and let q be an invertible element of R. The Hecke algebra Hn(q) = HR,q = HR,q(Sn) of the symmetric group Sn over R is defined as follows. As an R-module, Hn(q) is free with basis {gω| ω ∈ Sn}. The multiplication in Hn(q) satisfies the following relations:

(i) 1 ∈ Hn(q);

(ii) If ω = s1s2...sj is a reduced expression for ω ∈ Sn, then

gω = gs1 gs2 ...gsj ;

(iii) g2 = (q − 1)g + q for all transpositions s , where q = q.1 ∈ H (q). sj sj j n

−1 It is useful to abbreviate gsj by gj. Let R = Z[q, q ], and let n be a natural number. We use the term Hn(q) to indicate HR,q(Sn). Denote H2k+1,n(q) to be a subalgebra of the Hecke algebra which is generated by elements g2k+1, g2k+2, ..., gn−1 in Hn(q). As a free R-module, H2k+1,n(q) has an R-basis {gω| ω ∈ S2k+1,n}, In the next lemma we collect some basic facts on Hn(q).

0 0 0 Lemma 1.3.10. 1. If ω, ω ∈ Sn and l(ωω ) = l(ω) + l(ω ), then

gωgω0 = gωω0 .

2. Let sj be a transposition and ω ∈ Sn, then ( gsj ω if l(sjω) = l(ω) + 1 gjgω = (q − 1)gω + qgsj ω otherwise,

and ( gωsj if l(ωsj) = l(ω) + 1 gωgj = (q − 1)gω + qgωsj otherwise. 8 Background

3. Let ω ∈ Sn. Then gω is invertible in Hn(q) with inverse −1 −1 −1 −1 −1 gω = gj gj−1...g2 g1 , where ω = s1s2...sj is a reduced expression for ω, and

−1 −1 −1 −1 gj = q gj + (q − 1), so gj = qgj + (q − 1) for all sj.

In the literature, the Hecke algebras Hn(q) are equivalently deﬁned by generators gi, 1 ≤ i < n and relations

(H1) gigi+1gi = gi+1gigi+1 for 1 ≤ i ≤ n − 1;

(H2) gigj = gjgi for |i − j| > 1.

The next statement is implicit in [7] (or see Lemma 2.3 of [31]).

Lemma 1.3.11. The R-linear map i: Hn(q) −→ Hn(q) determined by i(gω) = gω−1 for each ω ∈ Sn is an involution on Hn(q)

Theorem 1.3.12. (Graham and Lehrer [15]) Let R = Z[q, q−1]. Then R-algebra Hn(q) is a cellular algebra. Remark 1.3.13. All facts above hold true for another version of the Hecke 2 2 2 algebra with parameter q and its subalgebra, say Hn(q ) and H2k+1,n(q ) 2 2 respectively. In detail, the Hecke algebra Hn(q ) (resp. H2k+1,n(q )) is deﬁned via the same generators and relations as for Hn(q) (resp. H2k+1,n(q)) respectively, but the parameter q is replaced by q2. These notions are going to be used from the Chapter 2.

The Murphy basis This section collects standard facts from the representation theory of the Hecke algebra. For using in Chapter 7, we recall the Murphy’s results 2 2 for the subalgebra H2k+1,n(q ) of Hn(q ) ; details can be found in [28] or [31]. If µ is a partition of n − 2k, deﬁne the element X cµ = gσ. (1.3.3)

σ∈Sµ

∗ For later convenience, we denote gω := i(gω) = gw−1 for ω ∈ S2k+1,n. Let ˇλ 2 H2k+1,n be the R-module in H2k+1,n(q ) with basis

∗ cst = gd(s)cµgd(t) : s, t ∈ Std(µ), where µ £ λ . (1.3.4)

2 Theorem 1.3.14. (Murphy [31]) The Hecke algebra H2k+1,n(q ) is free as an R–module with basis ∗ for s, t ∈ Std(λ) and M = cst = g cλgd(t) . (1.3.5) d(s) λ a partition of n − 2k Moreover, the following statements hold. 1.3. Hecke algebras of the symmetric groups 9

1. The R–linear involution i satisﬁes i : cst 7→ cts for all s, t ∈ Std(λ).

2 2. Suppose that h ∈ H2k+1,n(q ), and that s is a standard λ–tableau. Then there exist at ∈ R, for t ∈ Std(λ), such that for all s ∈ Std(λ),

X ˇλ csvh ≡ atcst mod H2k+1,n. (1.3.6) t∈Std(λ)

The basis M is cellular in the sense of [15]. If λ is a partition of λ 2 n − 2k, the cell (or Specht) module S for H2k+1,n(q ) is the R–module freely generated by

ˇλ {cs = cλgd(s) + H2k+1,n : s ∈ Std(λ)}, (1.3.7)

2 and with the right H2k+1,n(q )–action

X 2 csh = atct, for h ∈ H2k+1,n(q ), (1.3.8) t∈Std(λ)

where the coeﬃcients at ∈ R, for t ∈ Std(λ), are determined by the 2 expression (1.3.6). The basis M is called Murphy basis for H2k+1,n(q ) and the basis (1.3.7) is referred to as the Murphy basis for Sλ. Notice that the 2 λ H2k+1,n(q )–module S is dual to Specht module in [7]. Applying the general theory of cellular algebra, the bilinear form on Sλ is the unique symmetric R-bilinear map from Sλ × Sλ to R such that

ˇλ hcs, cticλ ≡ csi(ct) mod H2k+1,n (1.3.9)

for all s, t ∈ Std(λ). Then, rad Sλ = {x ∈ Sλ| hx, yi = 0 for all y ∈ Sλ} 2 λ is an H2k+1,n(q )-submodule of S . For each partition λ of n − 2k, denote λ λ λ 2 D = S /rad S a right H2k+1,n(q )-module. µ λ µ For partitions λ, µ of n − 2k and D 6= 0, let dλµ = [S : D ] be the composition multiplicity of Dµ in Sµ.

Theorem 1.3.15. (Dipper and James [7]) Suppose that R is a ﬁeld.

1. { Dµ — µ an e(q2)-restricted partition of n − 2k} is a complete set 2 of non-isomorphic simple H2k+1,n(q )–modules.

2. Suppose that µ is an e(q2)-restricted partition of n − 2k and that λ is

a partition of n − 2k. Then dµµ = 1 and dλµ 6= 0 only if λ ¤ µ.

Corollary 1.3.16. Suppose that R is a ﬁeld. Then the following statements are equivalent.

2 1. H2k+1,n(q ) is (split) semisimple; 10 Background

2. Sλ = Dλ for all partitions λ of n − 2k;

3. e(q2) > n − 2k.

Lemma 1.3.17. (Hemmer and Nakano [20], Proposition 4.2.1). Let e(q) ≥ 4, and suppose that µ £ λ. Then Ext1 (Sµ,Sλ) = 0. Hn(q) Lemma 1.3.18. (Hemmer and Nakano [20], Lemma 4.4.1). Let e(q) ≥ 3. Then

λ µ 1. If µ 6 ¤λ, then HomHn(q)(S ,S ) = 0.

λ λ 2. HomHn(q)(S ,S ) ≡ R.

1.4 Brauer algebra

Brauer algebras were introduced ﬁrst by Richard Brauer [2] in order to study the nth tensor power of the deﬁning representation of the orthogonal groups and symplectic groups. Afterwards, they were studied in more detail by various mathematicians. We refer the reader to work of Brown [3, 4], Hanlon and Wales [16, 17, 10], Graham and Lehrer [15], Koenig and Xi [25, 26], Wenzl [36] for more information.

Deﬁnition

The Brauer algebra is defined over the ring Z[x] via a basis given by diagrams with 2n vertices, arranged in two rows with n edges in each row, where each vertex belongs to exactly one edge. The edges which connect two vertices on the same row are called horizontal edges. The other ones are called vertical edges. We denote by Dn(x) the Brauer algebra where the vertices of diagrams are numbered 1 to n from left to right in both the top and the bottom row. Two diagrams d1 and d2 are multiplied by concatenation, that is, the bottom vertices of d1 are identified with the top vertices of d2, hence defining diagram d. Then d1 · d2 is defined γ(d1, d2) to be x d, where γ(d1, d2) denote the number of those connected components of the concatenation of d1 and d2 which do not appear in d, that is, which contain neither a top vertex of d1 nor a bottom vertex of d2. Let us demonstrate this by an example. We multiply two elements in D7(x):

• • •• • •• OO o OOO ooo d OOO oo 1 • • • • • ooOo • •

• • • •• •• ?? ooo ??ooo d2 •••ooo • ••• 1.4. Brauer algebra 11 and the resulting diagram is • • •• • •• 1 d1.d2 = x • •• • •••.

In ([2], Section 5) Brauer points out that each basis diagram on Dn(x) which has exactly 2k horizontal edges can be obtained in the form ω1e(k)ω2 where ω1 and ω2 are permutations in the symmetric group Sn, and e(k) is a diagram of the following form: • • ... • •• • ... • , • • ... • ••• ... • where each row has exactly k horizontal edges. As a consequence, the Brauer algebra can be considered over a polyno- mial ring over Z and is defined via generators and relations as follow: Take x to be an indeterminate over Z; Let R = Z[x] and define the Brauer algebra Dn(x) over R as the associative unital R–algebra generated by the transpositions s1, s2, . . . , sn−1, together with elements e(1), e(2), . . . , e([n/2]), which satisfy the defining relations:

2 (S0) si = 1 for 1 ≤ i < n;

(S1) sisi+1si = si+1sisi+1 for 1 ≤ i < n − 1;

(S2) sisj = sjsi for 2 ≤ |i − j|; i (1) e(k)e(i) = e(i)e(k) = x e(k) for 1 ≤ i ≤ k ≤ [n/2]; i−1 (2) e(i)s2je(k) = e(k)s2je(i) = x e(k) for 1 ≤ j ≤ i ≤ k ≤ [n/2];

(3) s2i+1e(k) = e(k)s2i+1 = e(k) for 0 ≤ i < k ≤ [n/2];

(4) e(k)si = sie(k) for 2k < i < n ;

(5) s(2i−1)s2ie(k) = s(2i+1)s2ie(k) for 1 ≤ i < k ≤ k ≤ [n/2];

(6) e(k)s2is(2i−1) = e(k)s2is(2i+1) for 1 ≤ i < k ≤ k ≤ [n/2];

(7) e(k+1) = e(1)s2,2k+1s1,2ke(k) for 1 ≤ k ≤ [n/2] − 1.

Regard the group ring RSn as the subring of Dn(x) generated by the transpositions

{si = (i, i + 1)for 1 ≤ i < n}.

Theorem 1.4.1. (Graham and Lehrer [15]) The Brauer algebra Dn(x) is cellular for any commutative noetherian integral domain R with identity and parameter x ∈ R. Deﬁnition 1.4.2. For any positive integer n, let X Z(n) = {r ∈ Z|r = 1 − c(p), µ ` k − 2, λ ` k, 2 ≤ k ≤ n}, p∈Y (λ\µ) p ∈ Y (λ/µ) where two boxes of Y (λ/µ) are not in the same column. 12 Background

Theorem 1.4.3. (Rui [35]) Let D(N) be the complex Brauer algebra.

1. Suppose N 6= 0. Then D(N) is semisimple if and only if N/∈ Z(n).

2. Dn(0) is semisimple if and only if n ∈ {1, 3, 5}. Theorem 1.4.4. (Rui [35])Let D(N) be the Brauer algebra over a ﬁeld R with charR > 0.

1. Suppose N 6= 0. Then D(N) is semisimple if and only if N/∈ Z(n) and charR - n!.

2. Dn(0) is semisimple if and only if n ∈ {1, 3, 5} and charR - n!. By Brown ([4], Section 3) the Brauer algebra has a decomposition as abelian groups [n/2] ∼ M Dn(x) = Z[x]Sne(k)Sn. k=0 Set M I(m) = Z[x]Sne(k)Sn, k≥m then I(m) is a two-sided ideal in Dn(x) for each m ≤ [n/2]. ∗ The modules Vk and Vk In this subsection we recall particular modules of Brauer algebras.

Dn(N) has a decomposition into direct some of vector spaces

[n/2] ∼ M Dn(N) = (Z[N]Sne(k)Sn + I(k+1))/I(k+1). k=0 Using the same arguments as in Section 1 ([39]), each factor module

(Z[N]Sne(k)ωj + I(k+1))/I(k+1) is a left Dn(N)-module with a basis given by the basis diagrams of Z[N]Sne(k)ωj, where ωj ∈ Sn is a diagram such that e(k)ωj is a diagram in Dn(N) with no intersection between any two vertical edges. In particular ∼ M I(k)/I(k+1) = (Z[N]Sne(k)ωj + I(k+1))/I(k+1), (1.4.1) j∈P (n,k) where P (n, k) is the set of all possibilities of ωj. As multiplication from the right by ωj commutes with the Dn(N)-action, each summand on the right hand side is isomorphic to the module

∗ Vk = (Z[N]Sne(k) + I(k+1))/I(k+1). (1.4.2) 1.4. Brauer algebra 13

∗ Combinatorially, Vk is spanned by basis diagrams with exactly k edges in the bottom row, where the i − th edge connects the vertices 2i − 1 and 2i. ∗ Observe that Vk is a free, ﬁnitely generated Z[N] module with Z[N]-rank n!/2kk!.

Similarly, a right Dn(N)-module is deﬁned

Vk = (Z[N]e(k)Sn + I(k+1))/I(k+1), (1.4.3)

∗ where basis diagrams are obtained from those in Vk by an involution, say ∗ *, of Dn(N) which rotates a diagram d ∈ Vk around its horizontal axis ∗ downward. For convenience later in Chapter 4 we use the term Vk replacing (k) ∗ Vn . More details for setting up Vk can be found in Section 1 of [39].

Lemma 1.4.5. (Wenzl [39], Lemma 1.1(d)) The algebra Dn(N) is L[n/2] ∗ L[n/2] faithfully represented on k=0 Vk (and also on k=0 Vk).

Length function for Brauer algebras Dn(N) Generalizing the length of elements in reﬂection groups, Wenzl [39] has deﬁned a length function for a diagram of Dn(N) as follows:

For a diagram d ∈ Dn(N) with exactly 2k horizontal edges, the deﬁnition of the length `(d) is given by

`(d) = min{`(ω1) + `(ω2)| ω1e(k)ω2 = d, ω1, ω2 ∈ Sn}.

We will call the diagrams d of the form ωe(k) where l(ω) = l(d) and ω ∈ Sn ∗ basis diagrams of the module Vk .

Remark 1.4.6. 1. Recall that the length of a permutation ω ∈ Sn is deﬁned by `(ω) = the cardinality of set

{(i, j)|(j)ω < (i)ω| 1 ≤ i < j ≤ n},

where the symmetric group acts on {1, 2, ..., n} on the right.

2. Given a diagram d, there can be more than one ω satisfying ωe(k) = d and `(ω) = `(d). e.g. s2j−1s2je(k) = s2j+1s2je(k) for 2j + 1 < k. This means that such an expression of d is not unique with respect to ω ∈ Sn.

For later use, if si = (i, i + 1) is a transposition in symmetric goup Sn with i, j = 1, ..., k let ( sisi+1...sj if i ≤ j, si,j = sisi−1...sj if i > j.

A permutation ω ∈ Sn can be written uniquely in the form

ω = tn−1tn−2...t1, where tj = 1 or tj = sij ,j with 1 ≤ ij ≤ j and 1 ≤ j < n. This can be seen as follows: For given ω ∈ Sn, there exists a unique tn−1 14 Background

0 −1 such that (n)tn−1 = (n)ω. Hence ω = (n)tn−1ω = n, and we can consider 0 ω as an element of Sn−1. Repeating this process on n implies the general claim. Set

∗ Bk = {tn−1tn−2...t2kt2k−2t2k−4...t2}. (1.4.4)

By the deﬁnition of tj given above, the number of possibilities of tj is j +1. ∗ k A direct computation shows that Bk has n!/2 k! elements. In fact, the ∗ ∗ number of elements in Bk is equal to the number of diagrams d in Dn(N) in which d∗ has k horizontal edges in each row and one of its rows is ﬁxed like that of e(k). 3. From now on, a permutation of a symmetric group is seen as a diagram with no horizontal edges, and the product ω1ω2 in Sn is seen as a concatenation of two diagrams in Dn(N). ∗ ∗ 4. Given a basis diagram d = ωe(k) with `(ω) = `(d ) does not imply ∗ 0 ∗ ∗ 0 that ω ∈ Bk, but there does exist ω ∈ Bk such that d = ω e(k). The latter will be shown in Lemma 1.4.8 to exist and to be unique for each basis ∗ ∗ 0 element of the module Vk . Wenzl even got `(d ) = `(ω) = `(ω ), where 0 0 ∗ `(ω ) is the number of factors for ω in Bk.

Example 1.4.7. To illustrate remark (2), we choose j = 1, k = 2. Given ∗ ∗ a basis diagram d in V2 is the following:

• • ••• • • d∗ = • •• ••••

• • • • • • • OOO OO s1s2 OOO • • O • • • • • = • •• •• • • e(2) • •• ••••

• • • • • • • ?? ?o?oo ??ooo ?? s3s2 • • ooo • • • • • = • •• •• • • e(2) • •• ••••

∗ ∗ In the picture d has two representations d = s1s2e(2) = s3s2e(2) satisfying

∗ `(d ) = `(s1s2) = `(s3s2) = 2. 1.4. Brauer algebra 15

∗ ∗ However, s1s2 is in B2 but s3s2 is not. In general, given a basis diagram d ∗ ∗ ∗ in Vk there always exists a unique permutation ω ∈ Bk such that d = ωek and l(d∗) = l(ω). The statement in the remark (4) above is shown in the following lemma.

Lemma 1.4.8. (Wenzl [39], Lemma 1.2)

∗ ∗ 1. The module Vk has a basis {ωv1 = vωe(k) , ω ∈ Bk} with `(ωe(k)) = `(ω). Here `(ω) is the number of factors for ω in (1.4.4), ∗ and v1 = (e(k) + I(k + 1))/I(k + 1) ∈ Vk .

∗ ∗ ∗ ∗ 2. For any basis element d of Vk , we have |`(sid )−`(d )| ≤ 1. Equality ∗ ∗ of lengths holds only if sid = d . For k ≤ [n/2], let

−1 ∗ Bk = {ω | ω ∈ Bk}. (1.4.5)

The following statement is similar to Lemma 1.4.8.

Lemma 1.4.9. 1. The module Vk has a basis {v1ω = ve(k)ω, ω ∈ Bk} with `(e(k)ω) = `(ω). Here `(ω) is the number of factors for ω in (1.4.5), and v1 = (e(k) + I(k + 1))/I(k + 1) ∈ Vk.

(k) 2. For any basis element d of Vn , we have |`(dsi) − `(d)| ≤ 1. Equality of lengths holds only if dsi = d. Chapter 2

The q-Brauer algebras

In this Chapter we review basic and necessary facts about the q-Brauer algebra due to Wenzl [39]. Then, we introduce more general versions for the q-Brauer algebra that are necessary for our work in this thesis. Apart from Wenzl’s results, Deﬁnitions 2.1.4 - 2.1.7 and Lemmas 2.1.10, 2.2.2 are new.

2.1 Definitions 1 − qN Definition 2.1.1. Fix N ∈ \{0} and let [N] = ∈ [q, q−1]. The Z 1 − q Z −1 q-Brauer algebra Brn(N) is defined over ring Z[q, q ] via generators g1, g2, g3, ..., gn−1 and e and relations

0 (H) The elements g1, g2, g3, ..., gn−1 satisfy the relations of the Hecke algebra Hn(q);

0 2 (E1) e = [N]e;

0 N −1 −1 (E2) egi = gie for i > 2, eg1 = g1e = qe, eg2e = q e and eg2 e = q e;

0 −1 −1 −1 −1 −1 −1 (E3) e(2) = g2g3g1 g2 e(2) = e(2)g2g3g1 g2 , where e(2) = e(g2g3g1 g2 )e.

A second version of the q-Brauer algebra is the following.

Deﬁnition 2.1.2. The q-Brauer algebra, denoted Brn(r, q), is deﬁned over ±1 ±1 the ring Z[q , r , (r − 1)/(q − 1)] by generators g1, g2, g3, ..., gn−1 and e and relations

(H) The elements g1, g2, g3, ..., gn−1 satisfy the relations of the Hecke algebra Hn(q); r − 1 (E ) e2 = e; 1 q − 1

16 2.1. Deﬁnitions 17

−1 −1 (E2) egi = gie for i > 2, eg1 = g1e = qe, eg2e = re and eg2 e = q e;

−1 −1 −1 −1 −1 −1 (E3) e(2) = g2g3g1 g2 e(2) = e(2)g2g3g1 g2 , where e(2) = e(g2g3g1 g2 )e. Remark 2.1.3. 1. A closely related algebra has appeared in Molev’s work. In 2003, Molev [29] introduced a new q-analogue of the Brauer algebra by considering the centralizer of the natural action in tensor spaces of a non- standard deformation of the universal enveloping algebra U(oN ). He defined relations for these algebras and constructed representations of them on tensor spaces. However, in general the representations are not faithful, and little is known about these abstract algebras besides their representations. The q-Brauer algebra, a closely related algebra with that of Molev, was introduced later by Wenzl [39] via generators and relations. In particular, over the field Q[r, q] he proved that it is semisimple and isomorphic to the Brauer algebra. It can be checked that the representations of Molev’s algebras in [29] are representations of q-Brauer algebras ([41], Section 2.2) and that the relations written down by Molev are satisfied by generators of the q-Brauer algebras; but potentially, Molev’s abstractly defined algebras could be larger ([30]). N 2. Obviously, by setting r = q the version Brn(r, q) coincides with Brn(N). Over a ring allowing to form the limit q → 1, such as the real or complex field, the q-Brauer algebra (both versions for q → 1) recovers the classical Brauer algebra Dn(N). In this case gi becomes the simple reflection si and the element e(k) can be identified with the diagram e(k). However, over any field of prime characteristic for which the limit q → 1 does not exist, Wenzl’s definitions cause technical difficulties to work. In particular, over a field of prime characteristic we can not give a comparison between the q-Brauer algebra and the classical Brauer algebra in the case q = 1 or q → 1. Further, if the coefficients [N] = 0 and (r − 1)/(q − 1) = 0, then we remark that the involution defined by Wenzl for the q-Brauer algebra (see [39], Remark 3.1.2) does not exist (a proof of this is in Lemma 3.2.1(3)). For studying in detail the q-Brauer algebra we subsequently give the modified versions for the q-Brauer algebra. So, the q-Brauer algebra can be considered over any field of characteristic p ≥ 0, as well as in the case q = 1 or q → 1.

Definition 2.1.4. Fix N ∈ Z \{0} and let [N] = 1 + q1 + ··· + qN−1. ±1 ±1 The q-Brauer algebra, Brn(N), over the ring Z[q , [N] ] is defined by the same generators and relations as in Definition 2.1.1.

Definition 2.1.5. Fix N ∈ Z \{0}, let q and r be invertible elements. Moreover, assume that if q = 1 then r = qN . The q-Brauer algebra ±1 ±1 ±1 Brn(r, q) over the ring Z[q , r , ((r − 1)/(q − 1)) ] is defined by the same generators and relations as in Definition 2.1.2. 18 The q-Brauer algebras

In Chapter 6 we need to use new versions of the q-Brauer algebra. In 2 these versions the q-Brauer algebra contains the Hecke algebra Hn(q ) as a subalgebra. The deﬁnitions are the following.

Deﬁnition 2.1.6. Let r and q be invertible elements over the ring Z[q±1, r±1, ((r − r−1)/(q − q−1))±1]. Moreover, if q = 1 then assume that N 2 2 r = q with N ∈ Z \{0}. The q-Brauer algebra Brn(r , q ) over the ring ±1 ±1 −1 −1 ±1 Z[q , r , ((r − r )/(q − q )) ] is the algebra deﬁned via generators g1, g2, g3, ..., gn−1 and e and relations

00 (H) The elements g1, g2, g3, ..., gn−1 satisfy the relations of the Hecke 2 algebra Hn(q );

r − r−1 (E )00 e2 = e; 1 q − q−1

00 2 −1 −1 (E2) egi = gie for i > 2, eg1 = g1e = q e, eg2e = rqe and eg2 e = (rq) e;

00 −1 −1 −1 −1 −1 −1 (E3) g2g3g1 g2 e(2) = e(2)g2g3g1 g2 , where e(2) = e(g2g3g1 g2 )e.

Definition 2.1.7. Fix N ∈ Z \{0} and let [N 2] = 1 + q2 + ··· + q2(N−1), where q is an invertible element in Z[q±1, r±1, ((r − r−1)/(q − q−1))±1]. The 2 ±1 ±1 −1 −1 ±1 q-Brauer algebra Brn(N ) over ring Z[q , r , ((r − r )/(q − q )) ] is defined by generators g1, g2, . . . , gn−1 and e and relations (H), (E3) as in Definition 2.1.6, and

0 2 2 (E1) e = [N ]e;

0 2 N+1 (E2) egi = gie for i > 2, eg1 = g1e = q e, eg2e = q e and −1 −1−N eg2 e = (q) e.

2 2 Remark 2.1.8. 1. In the q-Brauer algebra, the version Brn(r , q ) is 2 2 isomorphic with the version Brn(r, q). In fact, the version Brn(r , q ) can 2 2 −1 be obtained by substituting in Brn(r, q) old q, r and e by q , r and (q r)e respectively. 2. The new versions do not affect the properties of the q-Brauer algebra, which were studied in detail by Wenzl. This means that it is sufficient to give proofs of properties of the q-Brauer algebra for one version. Those of the other versions are the same. 3. In this thesis, we are going to work on Definitions 2.1.4 - 2.1.7 of the q-Brauer algebra replacing Wenzl’s. In particular, we start working with

the versions Brn(N) and Brn(r, q) from this Chapter, the other versions are intensively used in Chapter 6.

4. It is clear that in the case q = 1 the q-Brauer algebra Brn(N) (resp. 2 Brn(N )) coincides with the classical Brauer algebra Dn(N). And over a ring for which the limit q → 1 can be formed the q-Brauer algebra Brn(r, q) 2.1. Deﬁnitions 19

2 2 (resp. Brn(r , q )) recovers the classical Brauer algebra. Also note that all deﬁnitions above imply the equality

−1 −1 −1 −1 −1 −2 eg1 = g1 e = q e or eg1 = g1 e = q e. (2.1.1)

Let ( + glgl+1...gm if l ≤ m; gl,m = glgl−1...gm if l > m, and ( g−1g−1 ...g−1 if l ≤ m; g− = l l+1 m l,m −1 −1 −1 gl gl−1...gm if l > m, for 1 ≤ l, m ≤ n.

Deﬁnition 2.1.9. Let k be an integer, 1 ≤ k ≤ [n/2]. The element e(k) of the q-Brauer algebra is deﬁned inductively by e(1) = e and by

+ − e(k+1) = eg2,2k+1g1,2ke(k).

Notice that in this thesis we abuse notation by denoting e(k) both a certain diagram in the Brauer algebra Dn(N) and an element in the q-Brauer algebra Brn(r, q). Given a diagram d the geometric realization implies that diagrams e(k) and ω(d) commute on the Brauer algebra. Simi- larly, this also remains on the level of the q-Brauer algebra. The statement is the following.

Lemma 2.1.10. Let k be an integer, 1 ≤ k ≤ [n/2]. If gω ∈ H2k+1,n(q) 2 (resp. H2k+1,n(q )), then e(k)gω = gωe(k).

Proof. It is suﬃcient to show that e(k)gi = gie(k) with 2k + 1 ≤ i ≤ n − 1. This is shown by induction on k. Indeed, in the case k = 1, egi = gie for 3 ≤ i ≤ n − 1 by (E2). Suppose that

e(k−1)gi = gie(k−1) for 2k − 1 ≤ i ≤ n − 1.

Then for 2k + 1 ≤ i ≤ n − 1

Def2.1.9 + − Induction + − e(k)gi = (eg2,2k−1g1,2k−2e(k−1))gi = e(g2,2k−1g1,2k−2)gie(k−1) (H2) + − Def2.1.9 = gie(g2,2k−1g1,2k−2)e(k−1) = gie(k). 20 The q-Brauer algebras

2.2 Basic properties

Throughout this section, denote R to be a commutative ring containing ground rings in Deﬁnitions 2.1.4 - 2.1.7. The next lemmas indicate how the properties of the classical Brauer algebra extend to the q-Brauer algebra.

Lemma 2.2.1. Let Brn(N) be the q-Brauer algebra over R. Assume more that [N] and q are invertible elements in R. Then the following statements hold.

1. The elements e(k) are well-deﬁned.

+ + − − 2. g1,2le(k) = g2l+1,2e(k) and g1,2le(k) = g2l+1,2e(k) for l < k.

−1 −1 −1 −1 3. g2j−1g2je(k) = g2j+1g2je(k) and g2j−1g2j e(k) = g2j+1g2j e(k) for 1 ≤ j < k.

j 4. For any j ≤ k we have e(j)e(k) = e(k)e(j) = [N] e(k).

j−1 + − 5. [N] e(k+1) = e(j)g2j,2k+1g2j−1,2ke(k) for 1 ≤ j < k.

N j−1 6. e(j)g2je(k) = q [N] e(k) for 1 ≤ j ≤ k.

Lemma 2.2.2. Let Brn(r, q) be the q-Brauer algebra over R. Assume more that r, q, and (r − 1)/(q − 1) are invertible elements in R. Then the following statements hold. (1), (2), (3) as in Lemma 2.2.1. r − 1 4. For any j ≤ k we have e e = e e = ( )je . (j) (k) (k) (j) q − 1 (k) r − 1 5. ( )j−1e = e g+ g− e for 1 ≤ j < k. q − 1 (k+1) (j) 2j,2k+1 2j−1,2k (k) r − 1 6. e g e = r( )j−1e for 1 ≤ j ≤ k. (j) 2j (k) q − 1 (k) Proof. The proof is the same as that of Lemma 2.2.1 Lemma 2.2.3. We have P e(j)Hn(q)e(k) ⊂ H2j+1,n(q)e(k) + m≥k+1 Hn(q)e(m)Hn(q), where j ≤ k. Moreover, if j1 ≥ 2k and j2 ≥ 2k + 1, we also have: 1. eg+ g+ e = e g+ g− , if j ≥ 2k and j ≥ 2k + 1. 2,j2 1,j1 (k) (k+1) 2k+1,j2 2k+1,j1 1 2 2. eg+ g+ is equal to 2,j2 1,j1

k X e g+ g+ +qN+1(q−1) q2l−2(g +1)g+ g+ e . (k+1) 2k+2,j1 2k+1,j2 2l+1 2l+2,j2 2l+1,j1 (k) l=1 ∗ 2.3. The Brn(r, q)-modules Vk 21

P[n/2] Lemma 2.2.4. The algebra Brn(r, q) is spanned by k=0 Hn(q)e(k)Hn(q). In particular, its dimension is at most the one of the Brauer algebra.

Theorem 2.2.5. (restate a part of Theorem 5.3 in [39]) Let R be a

ﬁeld of characteristic zero. The algebra Brn(r, q) over R is semisimple if r 6= qk for |k| ≤ n and if e(q) > n (for e(q) see Deﬁnition 1.3.2). In this case, it has the same decomposition into simple matrix rings as the generic Brauer algebra, and the trace tr is nondegenerate.

∗ 2.3 The Brn(r, q)-modules Vk

∗ An action of generators of q-Brauer algebra on module Vk is deﬁned as follows:  qvd if sjd = d,  gjvd = vsj d if l(sjd) > l(d),  (q − 1)vd + qvsj d if l(sjd) < l(d), and

 N + − q hg v1 if g = 1,  3,j2 1,j1 ehg+ g− v = q−1hg− v if g− = 1, 2,j2 1,j1 1 j2+1,j1 1 1,j1  0 if j1 6= 2k and j2 6= 2k + 1, where v1 is deﬁned as in Lemma 1.4.8 and h ∈ H3,n.

∗ Lemma 2.3.1. The action of the elements gj with 1 ≤ j < n and e on Vk as given above deﬁnes a representation of Brn(r, q). Chapter 3

A basis and an involution for the q-Brauer algebra

In this chapter, we construct an explicit basis and provide an involution for the q-Brauer algebra. The ﬁnal section gives a comparison between this basis and another one due to Wenzl. These results are going to be used for producing cellular structure for the q-Brauer algebra over the commutative ring R in Chapter 4. Throughout, we prefer to work on the version Brn(r, q) of q-Brauer algebra. However, the other versions are still available.

3.1 A basis for q-Brauer algebra

By deﬁnition, the cellularity of an algebra depends on particular bases. That is, the cellular structure of an algebra can be only recognized on suitable bases. Here we give such a basis for the q-Brauer algebra. This basis is indexed by the set of all diagrams of the classical Brauer algebra Dn(N), where the parameter N is an integer N ∈ Z \{0}. Construction

Given a diagram d ∈ Dn(N) with exactly 2k horizontal edges, it is considered as concatenation of three diagrams (d1, ω(d), d2) as follows: 1. d1 is a diagram in which the top row has the positions of horizontal edges as these in the ﬁrst row of d, its bottom row is like a row of diagram e(k), and there is no crossing between any two vertical edges. 2. Similarly, d2 is a diagram where its bottom row is the same row in d, the other one is similar to that of e(k), and there is no crossing between any two vertical edges.

3. The diagram ω(d) is described as follows : We enumerate the free vertices, which belong to vertical edges, in both rows of d from left to right by 2k + 1, 2k + 2, ..., n. We also enumerate the vertices in each of two rows

22 3.1. A basis for q-Brauer algebra 23

of ω(d) from left to right by 1, 2, ..., 2k + 1, 2k + 2, ..., n. Assume that each vertical edge in d is connected by i − th vertex of the top row and j − th

vertex of the other one with 2k + 1 ≤ i, j ≤ n. Deﬁne ω(d) a diagram which has ﬁrst vertical edges 2k joining m−th points in each of two rows together with 1 ≤ m ≤ 2k, and its other vertical edges are obtained by maintaining the vertical edges (i, j) of d. Example 3.1.1. For n = 7, k = 2. Given a diagram • • • ••• • TTT eeeeeeeeee TTTT eeeeeeeeeee d = eTeTeTeTeeeeeee • eeee••eeeeeeeTT ••• • the diagram d can be expressed as product of diagrams in the following: • • • ••• • WWWW WWWWW WWWWW d1 • • • •WWWW • • •

• • • • • • • OO OOO ω . OOO (d) • • • • • • O •

• •• •• • • gggg gggg jjj gggggggggg jjj gggg gggg jjjj d2 •••gggggggggg •••jjj •

−2 Thus we have d = (N) · d1ω(d)d2.

Notice that diagram ω(d) can be seen as a permutation of symmetric group S2k+1,n. Since the expression above is unique with respect to each diagram d, the form (d1, ω(d), d2) is determined uniquely. By Lemmas ∗ 1.4.8 and 1.4.9, there exist unique permutations ω1 ∈ Bk and ω2 ∈ Bk such that d1 = ω1e(k) and d2 = e(k)ω2 with `(d1) = `(ω1), `(d2) = `(ω2). Thus, a diagram d is uniquely represented by the 3-tuple (ω1, ω(d), ω2) with ∗ −k ω1 ∈ Bk, ω2 ∈ Bk and ω(d) ∈ S2k+1,n such that d = N ω1e(k)ω(d)e(k)ω2 and `(d) = `(ω1) + `(ω(d)) + `(ω2). We call such a unique representation a reduced expression of d and brieﬂy write (ω1, ω(d), ω2).

Example 3.1.2. The example above implies that d1 = ω1e(2) with ∗ ω1 = s1,4s2 ∈ B2 , d2 = e(2)ω2 with ω2 = s4,1s5,2s6,4 ∈ B2 and ω(d) = s5s6. Thus we obtain

−2 d = N (ω1e(2))(s5s6)(e(2)ω2) = ω1e(2)s5s6ω2 = ω1s5s6e(2)ω2. Hence, d has a unique reduced expression

(ω1, s5s6, ω2) = (s1,4s2, s5s6, s4,1s5,2s6,4) with

`(d) = `(ω1) + `(s5s6) + `(ω2) = 5 + 2 + 11 = 18. 24 A basis and an involution for the q-Brauer algebra

Definition 3.1.3. For each diagram d of the Brauer algebra Dn(N), we define a corresponding element, say gd, in the q-Brauer algebra Brn(r, q) as follows: If d has exactly 2k horizontal edges and (ω1, ω(d), ω2) is a reduced expression of d, then define gd := gω1 e(k)gω(d) gω2 . If the diagram d has no horizontal edge, then d is seen as a permutation ω(d) of the symmetric group

Sn. In this case deﬁne gd = gω(d) . The main result of this section is stated below.

Theorem 3.1.4. The q-Brauer algebra Brn(r, q) over the ring R has a basis {gd |d ∈ Dn(N)} labeled by diagrams of the Brauer algebra.

Proof. A diagram d of Brauer algebra with exactly 2k horizontal edges has a unique reduced expression with data (ω1, ω(d), ω2). By the uniqueness of reduced expression with respect to a diagram d in D(N), the elements gd in Brn(r, q) are well-deﬁned. Observe that these elements gd belong to

Hn(q)e(k)Hn(q) since gω1 , gω(d) and gω2 are in Hn(q). Lemma 1.4.5 shows that there is a faithful representation of the Brauer algebra Dn(N) on L[n/2] ∗ k=0 Vk . By Lemma 2.3.1, this is a specialization of the representation of ∗ Brn(r, q) on the same direct sum of modules Vk , and hence, the dimension of Brn(r, q) has to be at least the one of Dn(N). Now, the other dimension inequality follows from using the result in Lemma 2.2.4. The theorem is proved.

3.2 An involution for the q-Brauer algebra

The following lemma provides more properties of the q-Brauer algebra.

+ − + − Lemma 3.2.1. 1. g2j+1g2,2k+1g1,2k = g2,2k+1g1,2kg2j−1, and − + − + g2k,1g2k+1,2g2j+1 = g2j−1g2k,1g2k+1,2 for 1 ≤ j ≤ k.

−1 −1 −1 2. g2j+1e(k) = e(k)g2j+1 = qe(k), and g2j+1e(k) = e(k)g2j+1 = q e(k) for 0 ≤ j < k.

− + 3. e(k+1) = e(k)g2k,1g2k+1,2e. Proof. 1. Let us to prove the ﬁrst equality, the other one is similar.

+ − (H2) + + − g2j+1g2,2k+1g1,2k = g2,2j−1(g2j+1g2jg2j+1)g2j+2,2k+1g1,2k (H1) + + − = g2,2j−1(g2jg2j+1g2j)g2j+2,2k+1g1,2k + + − = g2,2j+1g2jg2j+2,2k+1g1,2k (H2) + − (H2) + − − = g2,2k+1g2jg1,2k = g2,2k+1g1,2j−2g2jg2j−1,2k 3.2. An involution for the q-Brauer algebra 25

Lem1.3.10(3) + − −1 − = g2,2k+1g1,2j−2[(q − 1) + qg2j ]g2j−1,2k + − + − −1 −1 −1 − = (q − 1)g2,2k+1g1,2k + qg2,2k+1g1,2j−2(g2j g2j−1g2j )g2j+1,2k (H1) + − + − −1 −1 −1 − = (q − 1)g2,2k+1g1,2k + qg2,2k+1g1,2j−2(g2j−1g2j g2j−1)g2j+1,2k + − + − −1 − = (q − 1)g2,2k+1g1,2k + qg2,2k+1g1,2jg2j−1g2j+1,2k (H2) + − + − −1 = (q − 1)g2,2k+1g1,2k + qg2,2k+1g1,2kg2j−1 Lem1.3.10(3) + − + − −1 −1 = (q − 1)g2,2k+1g1,2k + qg2,2k+1g1,2k[q g2j−1 + (q − 1)] + − = g2,2k+1g1,2kg2j−1.

+ − Notice that when j = k then g2j+2,2k+1 = g2j+1,2k = 1, where 1 is the identity element in Brn(r, q). 2. To prove (2), we begin by showing the equality

g2j+1e(k) = qe(k) with 0 ≤ j < k. (3.2.1) Then the equality

−1 −1 g2j+1e(k) = q e(k) with 0 ≤ j < k (3.2.2) comes as a consequence. The other equalities will be shown simultaneously with proving (3). The equality (3.2.1) is shown by induction on k as follows:

For k = 1, the claim follows from (E2). Now suppose that the equality (3.2.1) holds for k − 1, that is,

g2j+1e(k−1) = qe(k−1) for j < k − 1. Then with j < k

(2.1.9) + − g2j+1e(k) = g2j+1eg2,2k−1g1,2k−2e(k−1) (a) for j

e(k)g2j+1 = qe(k) (3.2.3) and

−1 −1 e(k)g2j+1 = q e(k) for 0 ≤ j < k, (3.2.4) are proven by induction on k in a combination with (3) in the following way: If (3) holds for k − 1 then the equalities (3.2.3) and (3.2.4) are proven 26 A basis and an involution for the q-Brauer algebra

to hold for k. This result implies that (3) holds for k, and hence, the equalities (3.2.3) and (3.2.4), again, are true for k + 1. Proceeding in this way, all relations (3), (3.2.3) and (3.2.4) are obtained. Indeed, when k = 1 (3) follows from direct calculation:

(E3) −1 −1 e(2) = e(g2g3g1 g2 )e (3.2.5) Lem1.3.10(3) −1 −1 −1 −1 = e((q − 1) + qg2 )g3g1 (q g2 − q (q − 1))e q − 1 (q − 1)2 = eg g−1g e + eg−1g g−1g e − eg (g−1e) − (q − 1)eg−1g (g−1e) q 3 1 2 2 3 1 2 q 3 1 2 3 1 2 (E ) q − 1 (q − 1) =2 eg−1g g−1g e + g (eg−1)g e − eg (g−1e) − (q − 1)eg−1g (g−1e) 2 3 1 2 q 3 1 2 q 3 1 2 3 1 2 (2.1.1) q − 1 (q − 1) q − 1 = eg−1g g−1g e + g (eg e) − eg e − eg−1g e 2 3 1 2 q2 3 2 q2 3 q 2 3 2 (E ) (q − 1)r (q − 1) =2 eg−1g g−1g e + g e − e2g − q−1(q − 1)eg−1eg 2 3 1 2 q 3 q2 3 2 3 (E ), (E ) (q − 1)r (q − 1)(r − 1) q − 1 1 = 2 eg−1g g−1g e + g e − eg − eg 2 3 1 2 q2 3 q2 3 q2 3 −1 −1 (H2) −1 −1 = eg2 g3g1 g2e = eg2 g1 g3g2e. The above equality implies (3.2.3) for k = 2 and j < 2 in the following way:

(E3) + − (E2),(E3) e(2)g1 = eg2,3g1,2(eg1) = qe(2); (3.2.5) −1 −1 (E2) −1 −1 e(2)g3 = (eg2 g1 g3g2e)g3 = e(g2 g1 g3g2)g3e (1) for k=1 −1 −1 (E2) −1 −1 (3.2.5) = (eg1)g2 g1 g3g2e = qeg2 g1 g3g2e = qe(2). Therefore, in this case the equality (3.2.4) follows from multiplying the −1 equality (3.2.3) with g2j+1 on the right. As a consequence, (3) is shown to be true for k = 2 by following calculation.

− + (E3) + − − + (H2) + − − + − + e(2)g4,1g5,2e = (eg2,3g1,2e)g4,1g5,2e = (eg2,3g1,2e)(g4,3g5,4)g2,1g3,2e (E2) + − − + − + (H2),(E3) + −1 −1 = eg2,3g1,2(g4,3g5,4)(eg2,1g3,2e) = eg2,3g4 g5g1,3g4e(2) Lem1.3.10(3) + −1 −1 − −1 = eg2,3[q g4 + (q − 1)]g5g1,3[(q − 1) + qg4 ]e(2)

q − 1 q − 1 = eg+ g− e + eg+ g− e − eg+ g g− e − (q − 1)eg+ g g− e q 2,5 1,3 (2) 2,5 1,4 (2) q2 2,3 5 1,3 (2) 2,3 5 1,4 (2) 2 (2.1.9) q − 1 (q − 1) = eg+ g− e + e − eg+ g g− e − (q − 1)eg+ g g− e . q 2,5 1,3 (2) (3) q 2,3 5 1,3 (2) 2,3 5 1,4 (2) Subsequently, it remains to prove that

−1 + − −1 2 + − + − q (q −1)eg2,5g1,3e(2) −q (q −1) eg2,3g5g1,3e(2) −(q −1)eg2,3g5g1,4e(2) = 0. 3.2. An involution for the q-Brauer algebra 27

To this end, considering separately each summand in the left hand side of the last equality, it yields

q − 1 (H ) q − 1 eg+ g− e =2 eg+ g− g+ (g−1e ) (3.2.6) q 2,5 1,3 (2) q 2,3 1,2 4,5 3 (2) (3.2.2) for k=2 q − 1 = eg+ g− g+ e q2 2,3 1,2 4,5 (2) 2 Lem2.2.2(4) (q − 1) = (eg+ g− )g+ ee q2(r − 1) 2,3 1,2 4,5 (2) 2 (E ) (q − 1) =2 (eg+ g− e)g+ e q2(r − 1) 2,3 1,2 4,5 (2) 2 (E ) (q − 1) =3 e g+ e q2(r − 1) (2) 4,5 (2) 2 (E ), (H ) (q − 1) 2 = 2 e g e g q2(r − 1) (2) 4 (2) 5 3 Lem2.2.2(6) r(q − 1) = e g . q2(r − 1)2 (2) 5

2 2 (q − 1) (E ), (H ) (q − 1) eg+ g g− e 2 = 2 eg+ g− (g−1e )g (3.2.7) q 2,3 5 1,3 (2) q 2,3 1,2 3 (2) 5 2 (3.2.1) for k=2 (q − 1) = eg+ g− e g q2 2,3 1,2 (2) 5 3 Lem2.2.2(4) (q − 1) = (eg+ g− e)e g q2(r − 1) 2,3 1,2 (2) 5 (q − 1)3 = e e g q2(r − 1) (2) (2) 5 Lem2.2.2(4) (q − 1)(r − 1) = e g . q2 (2) 5

+ − (E2), (H2) + − − (q − 1)eg2,3g5g1,4e(2) = (q − 1)g5eg2,3g1,2g3,4e(2) (3.2.8) 2 Lem2.2.2(4) (q − 1) = g (eg+ g− )g− (ee ) r − 1 5 2,3 1,2 3,4 (2) 2 (E ) (q − 1) =2 g (eg+ g− e)g− e r − 1 5 2,3 1,2 3,4 (2) 2 (E ) (q − 1) =3 g (e g−1)g−1e r − 1 5 (2) 3 4 (2) 2 (3.2.4)for k=2 (q − 1) = g e g−1e q(r − 1) 5 (2) 4 (2) 2 Lem1.3.10(3) (q − 1) = g e [q−1g + (q−1 − 1)]e q(r − 1) 5 (2) 4 (2) 28 A basis and an involution for the q-Brauer algebra

(q − 1)2 (q − 1)3 = g e g e − g e e q2(r − 1) 5 (2) 4 (2) q2(r − 1) 5 (2) (2) 3 (E ),(H ),Lem2.2.2(4),(6) r(q − 1) (q − 1)(r − 1) 2 2 = e g − e g . q2(r − 1)2 (2) 5 q2 (2) 5

By (3.2.6), (3.2.7) and (3.2.8), it implies the equation (3) for k = 2. Now suppose that the relations (3.2.3) and (3.2.4) hold for k. We will show that (3) holds for k, and as a consequence both (3.2.3) and (3.2.4) hold with k + 1. Indeed, we have:

− + (2.1.9) + − −1 −1 − + e(k)g2k,1g2k+1,2e = (eg2,2k−1g1,2k−2e(k−1))(g2k g2k−1g2k−2,1)(g2k+1g2kg2k−1,2e) (H2) + − −1 −1 − + = (eg2,2k−1g1,2k−2e(k−1))(g2k g2k+1)(g2k−1g2k)g2k−2,1g2k−1,2e (E2), (H2) + − −1 −1 − + = eg2,2k−1g1,2k−2(g2k g2k+1)(g2k−1g2k)e(k−1)g2k−2,1g2k−1,2e (H2) + −1 − −1 − + = eg2,2k−1(g2k g2k+1)g1,2k−2(g2k−1g2k)(e(k−1)g2k−2,1g2k−1,2e).

By induction assumption, the last formula is equal to

+ −1 −1 − −1 eg2,2k−1[q g2k + (q − 1)]g2k+1g1,2k−1[(q − 1) + qg2k ]e(k), and direct calculation implies

+ − −1 + − eg2,2k+1g1,2ke(k) + q (q − 1)eg2,2k+1g1,2k−1e(k) −1 2 + − + − − q (q − 1) eg2,2k−1g2k+1g1,2k−1e(k) − (q − 1)eg2,2k−1g2k+1g1,2ke(k) (2.1.9) −1 + − = e(k+1) + q (q − 1)eg2,2k+1g1,2k−1e(k) −1 2 + − + − − q (q − 1) eg2,2k−1g2k+1g1,2k−1e(k) − (q − 1)eg2,2k−1g2k+1g1,2ke(k).

Applying the same arguments as in the case k = 3, each separate summand in the above formula can be computed as follows: q − 1 q − 1 eg+ g− e = eg+ g− (g−1 e ) (3.2.9) q 2,2k+1 1,2k−1 (k) q 2,2k+1 1,2k−1 2k−1 (k) (3.2.2) for k q − 1 (H ) q − 1 = eg+ g− e =2 eg+ g− g+ e q2 2,2k+1 1,2k−2 (k) q2 2,2k−1 1,2k−2 2k,2k+1 (k) k Lem2.2.2(4) (q − 1) = (eg+ g− )g+ (e e ) q2(r − 1)k−1 2,2k−1 1,2k−2 2k,2k+1 (k−1) (k) k (H ), (E ) (q − 1) 2 = 2 (eg+ g− e )g e g q2(r − 1)k−1 2,2k−1 1,2k−2 (k−1) 2k (k) 2k+1 k 2k−1 (2.1.9) (q − 1) Lem2.2.2(6) r(q − 1) = e g e g = e g . q2(r − 1)k−1 (k) 2k (k) 2k+1 q2(r − 1)2k−2 (k) 2k+1 3.2. An involution for the q-Brauer algebra 29

2 2 (q − 1) (H ), (E ) (q − 1) eg+ g g− e 2 = 2 g eg+ g− (g−1 e ) q 2,2k−1 2k+1 1,2k−1 (k) q 2k+1 2,2k−1 1,2k−2 2k−1 (k) (3.2.10) 2 (3.2.1) (q − 1) = g eg+ g− e q2 2k+1 2,2k−1 1,2k−2 (k) k+1 Lem2.2.2(4) (q − 1) = g (eg+ g− e )e q2(r − 1)k−1 2k+1 2,2k−1 1,2k−2 (k−1) (k) k+1 (2.1.9) (q − 1) = g e e q2(r − 1)k−1 2k+1 (k) (k) Lem2.2.2(4), (E ) (q − 1)(r − 1) = 2 e g q2 (k) 2k+1

+ − (H2),(E2) + − − (q − 1)eg2,2k−1g2k+1g1,2ke(k) = (q − 1)g2k+1(eg2,2k−1g1,2k−2)g2k−1,2ke(k) (3.2.11) k Lem2.2.2(4) (q − 1) = g (eg+ g− )g− (e e ) (r − 1)k−1 2k+1 2,2k−1 1,2k−2 2k−1,2k (k−1) (k) k (E ),(H ) (q − 1) 2 = 2 g (eg+ g− e )g− e (r − 1)k−1 2k+1 2,2k−1 1,2k−2 (k−1) 2k−1,2k (k) k (2.1.9) (q − 1) = g (e g−1 )g−1e (r − 1)k−1 2k+1 (k) 2k−1 2k (k) k (3.2.4) for k (q − 1) = g e g−1e q(r − 1)k−1 2k+1 (k) 2k (k) k Lem1.3.10(3) (q − 1) = g e [q−1g + (q−1 − 1)]e q(r − 1)k−1 2k+1 (k) 2k (k) (q − 1)k (q − 1)k+1 = g e g e − g (e )2 q2(r − 1)k−1 2k+1 (k) 2k (k) q2(r − 1)k−1 2k+1 (k) 2k−1 Lem2.2.2(4),(6) r(q − 1) (q − 1)(r − 1) = e g − e g . q2(r − 1)2k−2 (k) 2k+1 q2 (k) 2k+1

− + The above calculations imply that e(k)g2k,1g2k+1,2e = e(k+1). Thus (3) holds with value k. Using the relation (1), and the equalities (3.2.3) and (3) for k, it yields the equalities (3.2.3) and (3.2.4) for k + 1 as follows: For j < (k + 1) then

(3) for k − + e(k+1)g2j+1 = (e(k)g2k,1g2k+1,2e)g2j+1 (E2) − + (1)for k − + = e(k)(g2k,1g2k+1,2g2j+1)e = (e(k)g2j−1)g2k,1g2k+1,2e (3.2.3) for k − + (3) for k = q(e(k)g2k,1g2k+1,2e) = qe(k+1). The relation (3.2.4) for k + 1 is obtained by multiplying the relation (3.2.3) −1 for k + 1 by g2j+1 on the right. 30 A basis and an involution for the q-Brauer algebra

The next result provides an involution on the q-Brauer algebra

Brn(r, q).

Proposition 3.2.2. Let i be the map from Brn(r, q) to itself deﬁned by

i(gω) = gω−1 and i(e) = e for each ω ∈ Sn, extended an anti-homomorphism. Then i is an involution on the q-Brauer algebra Brn(r, q).

Proof. It is sufficient to show that i maps a basis element gd∗ to a basis ∗ element i(gd∗ ) on the q-Brauer algebra Brn(r, q). If given a diagram d ∗ with no horizontal edge, then d is as a permutation in Sn, it implies that obviously i(gd∗ ) = g(d∗)−1 = gd is a basis element of the q-Brauer algebra, where d is diagram which is obtained after rotating d∗ downward via an ∗ horizontal axis. If the diagram d = e(k), then by Definition 3.1.3 the corresponding basis element in the q-Brauer algebra is gd∗ = e(k). The equality i(gd∗ ) = i(e(k)) = e(k) is obtained by induction on k as follows: with k = 1 obviously i(e) = e by definition. Suppose i(e(k−1)) = e(k−1), then

(2.1.9) + − i(e(k)) = i(eg2,2k−1g1,2k−2e(k−1)) + − = i(e(k−1))i(g2,2k−1g1,2k−2)i(e) − + Lem3.2.1(3) = e(k−1)g2k−2,1g2k−1,2e = e(k).

∗ Now given a reduced expression (ω1, ω(d∗), ω2) of a diagram d , where ∗ ω1 ∈ Bk, ω2 ∈ Bk and ω(d∗) ∈ S2k+1,n, the corresponding basis element on ∗ the q-Brauer algebra Brn(r, q) is gd = gω1 gω(d∗) e(k)gω2 . This yields

∗ i(gd ) = i(gω1 gω(d∗) e(k)gω2 ) = i(gω2 )i(e(k))i(gω(d∗) )i(gω1 ) (3.2.12) Lem2.1.10 = g −1 e(k)g −1 g −1 = g −1 g −1 e(k)g −1 . ω2 ω(d∗) ω1 ω2 ω(d∗) ω1

∗ −1 −1 ∗ ω1 ∈ Bk and ω2 ∈ Bk imply that ω1 ∈ Bk and ω2 ∈ Bk. Therefore, −1 −1 −1 −1 by Lemmas 1.4.8 and 1.4.9, `(e(k)ω1 ) = `(ω1 ) and `(ω2 e(k)) = `(ω2 ). −1 −1 −1 This means that the 3-triple (ω2 , ω(d∗), ω1 ) is a reduced expression of the ∗ −k −1 −1 −1 diagram d = N ω2 e(k)ω ∗ e(k)ω1 . Thus i(gd∗ ) = g −1 g −1 e(k)g −1 is a (d ) ω2 ω(d∗) ω1 basis element in Brn(r, q) corresponding to the diagram d. The next corollary is needed for Chapter 4.

Corollary 3.2.3. The statements hold for the q-Brauer algebra Brn(r, q).

+ + − − 1. g2m−1,2je(k) = g2j+1,2me(k) and g2m−1,2je(k) = g2j+1,2me(k) for 1 ≤ m ≤ j < k.

+ + − − 2. e(k)g2l,1 = e(k)g2,2l+1 and e(k)g2l,1 = e(k)g2,2l+1 for l < k. 3.2. An involution for the q-Brauer algebra 31

+ + − − 3. e(k)g2j,2i−1 = e(k)g2i,2j+1 and e(k)g2j,2i−1 = e(k)g2i,2j+1 for 1 ≤ i ≤ j < k. r − 1 4. ( )j−1e = e g− g+ e for 1 ≤ j < k. q − 1 (k+1) (k) 2k,2j−1 2k+1,2j (j) r − 1 5. e g e = r( )j−1e for 1 ≤ j ≤ k. (k) 2j (j) q − 1 (k) P 6. e(k)Hn(q)e(j) ⊂ e(k)H2j+1,n(q) + m≥k+1 Hn(q)e(m)Hn(q), where j ≤ k. Proof. These results, without the relations (1) and (3), are directly deduced from Lemmas 2.2.2 and 2.2.3 using the property of the above involution. The statement (1) can be proven by induction on m as follow: With m = 1 obviously (1) follows from Lemma 2.2.2(2) with j = l. Suppose that (1) holds for m − 1, that is,

+ + g2m−3,2(j−1)e(k) = g2(j−1)+1,2m−2e(k) and (3.2.13)

− − g2m−3,2(j−1)e(k) = g2(j−1)+1,2m−2e(k) for 1 ≤ m ≤ j < k. (3.2.14) Then

+ − + + g2m−1,2je(k) = g2m−2,2m−3(g2m−3,2j−2)g2j−1,2je(k) Lem2.2.2(3) − + + = g2m−2,2m−3(g2m−3,2j−2)g2j+1,2je(k) (H2) − + + = g2m−2,2m−3g2j+1,2j(g2m−3,2j−2)e(k) (3.2.13) − + + = g2m−2,2m−3g2j+1,2j(g2j−1,2m−2)e(k) − + = g2m−2,2m−3g2j+1,2m−2e(k) (H2) + − + = g2j+1,2mg2m−2,2m−3g2m−1,2m−2e(k) Lem2.2.2(3) + − + = g2j+1,2mg2m−2,2m−3g2m−3,2m−2e(k) + = g2j+1,2me(k). The other equality is proven similarly. The relation (3) is directly deduced from (1) by using the involution. Notice that the equality (3) in Lemma 2.2.2 is the special case of the above equality (1). Remark 3.2.4. In the proof of Lemma 3.2.1, the properties (2) and (3) are shown by using assumption (r−1)/(q−1) invertible. So, over commutative r − 1 ring R that (r − 1)/(q − 1) is not invertible, such as R = [q±1, r±1, ] Z q − 1 in Deﬁnition 2.1.2, these properties do not hold true. This implies that Proposition 3.2.2 is wrong. Thus, the map i is not an involution on the q-Brauer algebra Brn(r, q) if (r − 1)/(q − 1) is not invertible. 32 A basis and an involution for the q-Brauer algebra

3.3 An algorithm producing basis elements

We introduce here an algorithm which produces basis elements gd for the q-Brauer algebra Brn(r, q) from a given diagram d in the classical Brauer algebra Dn(N). This algorithm’s construction is based on the proof of Lemma 1.4.8(1) (see [39], Lemma 1.2(a) for a complete proof). From the expression of an arbitrary diagram d as concatenation of three partial ∗ diagrams (d1, ω(d), d2) in Section 3.1 it is suﬃcient to consider diagrams d ∗ as the form of d1. That is, d has exactly k horizontal edges on each row, its bottom row is like a row of the diagram e(k), and there is no crossing ∗ ∗ between any two vertical edges. Let Dk,n be the set of all diagrams d above.

Recall that a permutation si,j with 1 ≤ i, j ≤ n − 1 can be considered ∗ ∗ as a diagram, say d(i,j+1) if i ≤ j or d(i+1,j) if i > j, in the Brauer algebra such that its free points, including 1, 2,...,i − 1, j + 2, ... n, are ﬁxed.

Example 3.3.1. In D7(N) the permutation s6,3 corresponds to the following diagram • • • • • • • ∗ ? ? ? g?gg d = ? ? g?gggg ? (7,3) ?? gg?g?gg ?? ?? • ••••••ggggg

The algorithm ∗ ∗ ∗ Given a diagram d of Dk,n, we number the vertices in both rows of d from left to right by 1, 2, ..., n. Note that for 2k + 1 ≤ i ≤ n if the i-th vertex in its bottom row joins to the f(i)-th vertex in the top row, then f(i) < f(i+1) since there is no intersection between any two vertical edges ∗ ∗ in the diagram d . This implies that concatenation of diagrams d(n,f(n)) and d∗ yields a new diagram

∗ ∗ ∗ d1 = d(n,f(n))d whose n-th vertex in the bottom row joins that of the top row and whose ∗ ∗ other vertical edges retain those of d . That is, the diagram d1 has the (n − 1)-th vertex in its bottom row joining to the point f(n − 1)-th vertex ∗ ∗ in the top row. Again, a concatenation of diagrams d(n−1,f(n−1)) and d1 produces a diagram

∗ ∗ ∗ ∗ ∗ ∗ d2 = d(n−1,f(n−1))d1 = d(n−1,f(n−1))d(n,f(n))d whose n-th and (n − 1)-th vertices in the bottom row join, respectively, those of the top row and whose other vertical edges maintain these in d∗. ∗ Proceeding in this way, we determine a series of diagrams d(n,f(n)), ∗ ∗ d(n−1,f(n−1)),. . . , d(2k+1,f(2k+1)) such that 0 ∗ ∗ ∗ ∗ d = d(2k+1,f(2k+1))...d(n−1,f(n−1))d(n,f(n)) d 3.3. An algorithm producing basis elements 33

0 is a diagram in S2ke(k). Here, d can be seen as a diagram in D2k(N) with only horizontal edges to which we add (n − 2k) strictly vertical edges to the right. Subsequently, set i2k−2 the label of the vertex in the top row of d0 which is connected with the 2k-th vertex in the same row and 0 −1 0 t(2k−2) = si2k−2,2k−2. Then the new diagram d1 = t(2k−2)d has two vertices 2k-th and (2k − 1)-st in the top row which are connected by a horizontal ∗ edge. Proceeding this process, ﬁnally d transforms into e(k). −1 −1 −1 ∗ ∗ ∗ ∗ e(k) = t(2)...t(2k−4)t(2k−2)d(2k+1,f(2k+1))...d(n−1,f(n−1))d(n,f(n)) d . ∗ ∗ Hence, d can be rewritten as d = ωe(k), where

−1 −1 −1 ∗ ∗ ∗ −1 ω = (t(2)...t(2k−4)t(2k−2)d(2k+1,f(2k+1))...d(n−1,f(n−1))d(n,f(n))) (3.3.1) ∗ ∗ ∗ −1 −1 −1 −1 −1 = (d(2k+1,f(2k+1))...d(n−1,f(n−1))d(n,f(n))) (t(2)...t(2k−4)t(2k−2)) ∗ ∗ ∗ = d(f(n),n)d(f(n−1),n−1)... d(f(2k+1),2k+1)t(2k−2)t(2k−4)...t(2)

= sf(n),n−1sf(n−1),n−2... sf(2k+1),2ksi2k−2,2k−2si2k−4,2k−4... si2,2 with f(i) < f(i + 1) for 2k + 1 ≤ i ≤ n − 1. ∗ Notice that the involution ∗ in Dn(N) maps d to the diagram d which −1 is of the form e(k)ω with −1 ω = s2,i2 ... s2k−4,i2k−4 s2k−2,i2k−2 s2k,f(2k+1)... sn−2,f(n−1)sn−1,f(n).

By Deﬁnition 3.1.3, the corresponding basis elements gd∗ and gd in the q -Brauer algebra are

gd∗ = gωe(k) and gd = e(k)gω−1 .

∗ Example 3.3.2. In D7(N) we consider the following diagram d • • • • • •• WWWW WWWW WWWW ∗ WWWWWWWWWWWWWWW . d WWWWWWWWWWWWWWW • •• ••••WWWW WWWW WWWW Step 1. Transform d into d0 • • • • • • • ∗ ? ? ? g?gg d ? ? g?gggg ? (7, 3) ?? gg?g?gg ?? ?? • • • gggg•g • • • • • • • ••• WWWW WWWW WWWWWWWWWW ∗ ; = WWWWWWWWWW d1 • • • • • ••• •• ••••WWWW WWWW WWWW WWWW WWWW ∗ WWWWWWWWWWWWWWW d WWWWWWWWWWWWWWW • •• ••••WWWW WWWW WWWW

• • • • • • • ∗ ? ? ? g?gg d ? ? g?gggg ? (6, 2) ?? gg?g?gg ?? ?? • • gggg•g • • • • • • • ••• • WWWW WWWWW ∗ ; = WWWWW d2 • • • • ••• • •• ••••WWWW WWWW WWWW ∗ WWWWWWWWWW d1 WWWWWWWWWW • •• ••••WWWW WWWW 34 A basis and an involution for the q-Brauer algebra

• • • • • • • ∗ ? ? ? g?gg d ? ? g?gggg ? (5, 1) ?? gg?g?gg ?? ?? • gggg•g • • • • • • • ••• • • = d0 ; • • • ••• • • •• •••• WWWW ∗ WWWWW d2 WWWWW • •• ••••WWWW

0 Step 2. Transform d into e(2)

• • • • • • • ?? t(2) ?? • • • • • • • • •• •• • • = e(2). • • ••• • • • •• •••• d0 • •• ••••

∗ Now, the diagram d is rewritten in the form ωe(2), where

∗ ∗ ∗ −1 ω = (d(5, 1)d(6, 2)d(7, 3)) t(2) = s3,6s2,5s1,4s2.

∗ The corresponding basis element with d in the q-Brauer algebra Br7(r, q) is + + + gd∗ = gωe(2) = g3,6g2,5g1,4g2e(2).

Using the involution ∗ in the Brauer algebra Dn(N) yields the resulting diagram d in which

−1 −1 d = e(2)ω = e(2)(s3,6s2,5s1,4s2) = e(2)s2s4,1s5,2s6,3.

Hence, + + + gd = e(2)gω−1 = e(2)g2g4,1g5,2g6,3. Observe that this result can also be obtained via applying the involution i (see Proposition 3.2.2) on the q-Brauer algebra, that is,

+ + + + + + i(gd∗ ) = i(gωe(2)) = i(g3,6g2,5g1,4g2e(2)) = e(2)g2g4,1g5,2g6,3 = gd. Remark 3.3.3. 1. Combining with Lemma 1.4.8 the algorithm above ∗ ∗ implies that given a diagram d of Dk,n there exists a unique element ∗ ω = tn−1tn−2...t2kt2k−2t2k−4...t2 ∈ Bk with tj = sij ,j and ij < ij+1 for ∗ ∗ 2k + 1 ≤ j ≤ n − 1, such that d = ωe(k) and `(d ) = `(ω). Let

∗ ∗ ∗ ∗ ∗ ∗ Bk,n = {ω ∈ Bk | d = ωe(k) and `(d ) = `(ω), d ∈ Dk,n} (3.3.2) and

−1 ∗ Bk,n = {ω | ω ∈ Bk,n }. (3.3.3) 3.4. A comparison 35

By Lemma 1.4.9(1),

−1 −1 −1 Bk,n = {ω ∈ Bk| d = e(k)ω and `(d) = `(ω ), d ∈ Dk,n},

∗ ∗ where Dk,n is the set of all diagrams d which are image of d ∈ Dk,n via the involution ∗. The uniqueness of element ω ∈ Bk,n means that

∗ ∗ |Bk,n| = |Bk,n| = |Dk,n| = |Dk,n|.

∗ ∗ ∗ Given a diagram d in Dk,n, since the number of diagrams d is equal to the number of possibilities to draw k edges between n vertices on its top row, it implies that

n! |B∗ | = |B | = . k,n k,n 2k(n − 2k)!k!

∗ 2. For an element ω = tn−1tn−2...t2kt2k−2t2k−4...t2 ∈ Bk,n if tj = 1 with 2k ≤ j ≤ n − 1 then tj+1 = 1. Indeed, as in (3.3.1), suppose that ∗ tj = sf(j+1),j = 1. This means the corresponding diagram d(f(j+1),j+1) = 1, that is a diagram with all vertical edges and no intersection between any two vertical edges. It implies that the (j +1)-st vertex in the bottom row of diagram d∗ joins the same vertex of the upper one, that is f(j + 1) = j + 1. ∗ ∗ By deﬁnition of d in Dk,n the other vertical edges on the right side of the (j + 1)-st vertex of the bottom row has no intersection. This means the (j+2)-th vertex in the bottom row of diagram d∗ joins the f(j+2) = (j+2) - ∗ ∗ th vertex of the top row. Hence, d(f(j+2),j+2) = d(j+2,j+2) = 1, that is tj+1 = sf(j+2),j+1 = sj+2,j+1 = 1.

3.4 A comparison

Wenzl introduced a reduced expression of Brauer diagram d in the general way

`(d) = min{`(δ1) + `(δ2): d = δ1e(k)δ2, δ1, δ2 ∈ Sn}. This definition allows for the existence of several different reduced expressions with respect to a diagram d. If a reduced expression d = δ1e(k)δ2, δ1, δ2 ∈ Sn is fixed, then he can define a basis element gd = gδ1 e(k)gδ1 for the q-Brauer algebra Brn(r, q) (see [39], Section 3.7). In Section 3.1 we have given the notation of reduced expression of a diagram d with a different way. Our construction also is based on the length function of diagram d as Wenzl’s. However, here the diagram d is −k presented by partial diagrams d1, d2 and ω(d) such that d = N d1ω(d)d2. This allows us to produce a unique reduced expression of d. Hence, we can give another basis for the q-Brauer algebra. 36 A basis and an involution for the q-Brauer algebra

No surprise that the length of a diagram d is the same in both our description and that of Wenzl, since suppose that d has a reduced expression

(ω1, ω(d), ω2). Then

−k −k d = N d1ω(d)d2 = N (ω1e(k))ω(d)(e(k)ω2) −k 2 = N (ω1ω(d))e(k)ω2 = (ω1ω(d))e(k)ω2 = ω1e(k)(ω(d)ω2)

By Wenzl’s deﬁnition d has another reduced expression d = δ1e(k)δ2. This implies that

d = (ω1ω(d))e(k)ω2 = ω1e(k)(ω(d)ω2) = δ1e(k)δ2.

Hence, (δ1, δ2) = (ω1ω(d), ω2) or (δ1, δ2) = (ω1, ω(d)ω2). Thus

`(d) = `(ω1) + `(ω(d)) + `(ω2) = min{`(δ1) + `(δ2)}. Remark 3.4.1. 1. We remark that the basis due to Wenzl is not suitable for pointing out the cellular structure of the q-Brauer algebra (see Example 3.4.3). 2. The basis constructed in Section 3.1 can produce cellularity for the q-Brauer algebra. This is shown in the next chapter. Example 3.4.2. This example demonstrates that there are two different bases for the q-Brauer algebra following Wenzl’s Definition, but there is a unique basis in our construction We consider the diagram d as in Example 3.1.1. Using reduced expression definition of Wenzl, d can be determined as a product

d = δ1e(k)δ2 in the following ways: Case 1: d has a reduced expression d = δ1e(k)δ2 such that e(k)δ2 is an diagram which has no intersection between any two vertical edges.

• • • • • • • ZZZZZZZ o ZZoZoZoZZZ ooo ZZZZZZ δ1 • • oo • • • ZZZ•ZZZZ •

• • • ••• • • •• •• • • TTTT eeeeeeeeeee TTT eeeeeeeeee = e(2). d eTeTeTeTeeeeeee • eeee••eeeeeeeTT ••• •• • • • • • •

• • T • T • T • • • OO TTT TTgTgggTgTgTgggg jjj OOO TgTgTgg gTgTgTg TTTjjj gOgOggg ggTgTgTT TTjTjTjjTTTT δ2 •••••••gggg ggOggg jTjTj TT TT

∗ Where δ1 = ω1s5s6 = s1,4s2s5s6 = s1,6s2 ∈ B2 with `(δ1e(2)) = `(δ1) = 7, and δ2 = ω2 = s4,1s5,2s6,4 ∈ B2, with `(e(2)δ2) = `(δ2) = 11. In this case d has a reduced expression

(δ1, δ2) = (s1,6s2, s4,1s5,2s6,4) 3.4. A comparison 37 with

`(d) = `(δ1) + `(δ2) = 7 + 11 = 18. The corresponding basis element in the q-Brauer algebra following Wenzl’s deﬁnition is

+ + + + gd = gδ1 e(2)gδ2 = g1,6g2e(2)g4,1g5,2g6,4. (3.4.1)

Case 2: d has a reduced expression d = δ1e(k)δ2 such that δ1e(k) is an diagram which has no intersection between any two vertical edges.

• • • • • • • WWWW o WWWWWooo ooWWWWW δ1 • • ooo • •WWWW • • •

• • • ••• • • •• •• • • TTTT eeeeeeeeeee TTT eeeeeeeeee = e(2) . d eTeTeTeTeeeeeee • eeee••eeeeeeeTT ••• •• • • • • • •

• • • • • • • OO TTTT TTTT TTeTeTeeeeeeeee OOO TTTeeeTeTeeTeeeTeTeT OeOeeeeeeTeTeTeTeTTTT TTTT δ2 •••••••eeeeeeeOeeee TT TT TT

∗ Where δ1 = ω1 = s1,4s2 ∈ B2 with `(δ1e(2)) = `(δ1) = 5, and δ2 = s5s6ω2 = s5s6s4,1s5,2s6,4 = s4,2s5,1s6,2 ∈ B2 with `(e(2)δ2) = `(δ2) = 13. The reduced expression of d in this case is

(δ1, δ2) = (s1,4s2, s4,2s5,1s6,2) with

`(d) = `(δ1) + `(δ2) = 5 + 13 = 18. The corresponding basis element in the q-Brauer algebra following Wenzl’s deﬁnition is

+ + + + gd = gδ1 e(2)gδ2 = g1,4g2e(2)g4,2g5,1g6,2. (3.4.2)

Two cases above yield that Wenzl’s deﬁnition of the reduced expression of a diagram d is not unique since

(s1,4s2, s4,1s5,2s6,4) 6= (s1,6s2, s4,2s5,1s6,2).

Comparing with Example 3.1.2 it implies that `(d) is the same, but the corresponding basis element with d for the q-Brauer algebra, following Deﬁnition 3.1.3, is

+ + + + gd = gω1 e(2)gω(d)gω2 = g1,4g2(g5g6)e(2)g4,1g5,2g6,4. (3.4.3) 38 A basis and an involution for the q-Brauer algebra

Example 3.4.3. This example illustrates that the bases due to Wenzl are not suitable for providing the cellular structure of the q-Brauer algebra. To show this, we apply the condition (C2) in Deﬁnition 1.1.1 of the cellular algebra to basis elements of the q-Brauer algebra calculated in the

last example. A basis {gd} is called good if it satisﬁes (C2). Note that the involution i of the q-Brauer algebra was deﬁned in Proposition 3.2.2. We consider two cases in Example 3.4.2.

Case 1: The involution i maps the basis element gd in the equality (3.4.1) to the element

+ + + + i(gd) = i(gδ1 e(2)gδ2 ) = i(g1,6g2e(2)g4,1g5,2g6,4) (3.4.4) + + + + = g −1 e(2)g −1 = g g g e(2)g2g δ2 δ1 4,6 2,5 1,4 6,1

Using the Wenzl’s deﬁntition the basis element i(gd) in the q-Brauer algebra yields a corresponding Brauer diagram

0 −1 −1 d = s4,6s2,5s1,4e(2)s2s6,1 = δ2 e(2)δ1 . The concatenation of component Brauer diagrams implies that d0 is an image of d reﬂected via a horizontal axis.

• • W • • • • • WWWWW WoWWW TjTjT jj jj ooWoWWW WWjWjWjj jjTjTjTTjjjj −1 oo jWjWjWWWWjWjWjWWjjjTTT δ2 • oo • jjjj• jjjj•WWjWjWjj•WWWW •TTT •

• • • • •• • • •• •• • • YYYYYYYYYY jj 0 YYYjYjYjYYYYYY e . d jjjj YYYYYYYYYYY = (2) • jjjj• • •••••YYYYYYYYYYY • • • • • •

• • • • • • • ? OO ? ?ddddd?dd ?? OOO dd??dddd?? ?? −1 ? ddOdOdOddd ? ? ? δ1 •••••••ddddddd O Applying the algorithm in Section 3.3, it is clear that the expression

0 −1 −1 d = δ2 e(2)δ1 = s4,6s2,5s1,4e(2)s2s6,1 is a reduced expression of d0 following Wenzl’s deﬁnition. Hence, the ele-

ment gd0 = i(gd) is a basis element of the q-Brauer algebra. However, by a −1 geometry realization the diagram δ2 e(2) has no intersection between any −1 two vertical edges, but e(2)δ1 totally has two intersections. So, the basis element i(gd) is not in the case 1 but in the case 2 of Example 3.4.2. This means the involution i maps the basis element gd in a basis to the basis element i(gd) of another basis. In particular, if call B1 and B2 are bases of the q-Brauer algebra shown in two cases 1 and 2 respectively. Then

gd ∈ B1, but i(gd) ∈/ B1 and i(gd) ∈ B2. Thus, the basis B1 of the q-Brauer algebra does not satisfy the condition (C2). Case 2. Using the same arguments as in Case 1, we can remark that

the basis B2 of the q-Brauer algebra does not satisfy (C2). Chapter 4

Cellular structure of the q-Brauer algebra

This section is devoted to establish cellularity of the q-Brauer algebra on the version Brn(r, q). All results in this section hold true for the other versions. In fact, to describe cellular structure of Brn(r, q) we need to use an explicit basis which has been introduced in the previous chapter. The first main result in this chapter is Theorem 4.2.2 that states: The q-Brauer algebra over commutative ring is a cellular algebra with respect to the involution i. Further, it is an iterated inflation of Hecke algebras of the symmetric groups. More detail about the concept ”iterated inflation” is given in [24, 26]. The second main result is Theorem 4.2.7 which gives a complete set of indexes of simple modules, up to isomorphism, of the q-Brauer algebra.

4.1 An iterated inﬂation

∗ The free ﬁnitely generated R-module Vk,n and Vk,n

As usual let k be an integer, 0 ≤ k ≤ [n/2], and denote Dn(N) the ∗ classical Brauer algebra over commutative ring R with unity. Deﬁne Vk,n ∗ ∗ to be an R-vector space linearly spanned by Dk,n. This implies Vk,n is an ∗ R-submodule of Vk . Hence, by Lemma 1.4.8(1) a given basis diagram ∗ ∗ ∗ d in Vk,n has a unique reduced expression in the form d = ωe(k) with ∗ ∗ `(d ) = `(ω) and ω ∈ Bk,n. Similarly, let Vk,n be an R-vector space linearly spanned by Dk,n. That is, a basis diagram d in Vk,n has exactly 2k horizontal edges, its top row is the same as a row of e(k), and there is no intersection between any two vertical edges.

The following lemma is directly deduced from deﬁnitions.

39 40 Cellular structure of the q-Brauer algebra

∗ ∗ ∗ Lemma 4.1.1. The R-module Vk,n has a basis {d = ωe(k), ω ∈ Bk,n}. Dually, the R-module Vk,n has a basis {d = e(k)ω, ω ∈ Bk,n}. Moreover, n! dim V = dim V ∗ = . R k,n R k,n 2k(n − 2k)!k!

Statements below in Lemmas 4.1.2, 4.1.4, 4.1.5 and Corollary 4.1.3 are needed for later reference.

∗ ∗ Lemma 4.1.2. Let d be a basis diagram in Vk,n and π be a permutation in ∗ ∗ ∗ ∗ S2k+1,n. Then d π is a basis diagram in Vk satisfying `(d π) = `(d )+`(π). ∗ ∗ Proof. Since d ∈ Vk,n, it implies that there exists a unique element ∗ ∗ ω ∈ Bk,n such that d = ωe(k). Consider π as a diagram in Dn(N) and also observe that two diagrams e(k) and π commute. It follows from concatenation of diagrams d∗ and π that the diagram d∗π is a basis diagram ∗ ∗ ∗ in Vk . This yields d π = ωe(k)π = ωπe(k). Since the basis diagram d has no intersection between any two vertical edges, the number of intersections of vertical edges in the resulting basis diagram d∗π is equal to that of the diagram π. In fact, the number of intersections of vertical edges in the diagram π is equal to its length. This produces `(d∗π) = `(d∗) + `(π).

∗ Corollary 4.1.3. Let ω be a permutation in Bk,n and π be a permutation in S2k+1,n. Then

∗ `(ωπ) = `(ω) + `(π) and ωπ ∈ Bk.

∗ ∗ ∗ Proof. Set d = ωe(k), then `(d ) = `(ω) since ω ∈ Bk,n. Lemma 4.1.2 ∗ ∗ ∗ implies that `(d π) = `(d )+`(π) = `(ω)+`(π) and d π = ωπe(k) is a basis ∗ ∗ diagram in Vk . The latter deduces that `(d π) = `(ωπe(k)) ≤ `(ωπ), that is, `(d∗π) ≤ `(ωπ). Immediately, the ﬁrst equality follows from applying the inequality `(ω) + `(π) ≥ `(ωπ). 0 ∗ ∗ Let d = d π ∈ Vk , then by Lemma 1.4.8 there uniquely exists an 0 ∗ 0 0 0 0 element ω in Bk such that d = ω e(k) and `(d ) = `(ω ). This implies that

0 0 d = ωπe(k) = ω e(k) (4.1.1) and `(d0) = `(ωπ) = `(ω) + `(π) = `(ω0). −1 0 The equality (4.1.1) yields πe(k) = ω ω e(k). Observe that the basis diagram πe(k) is a combination of two separate parts where the ﬁrst part includes all horizontal edges on the left side and the second consists of all

vertical edges on the other side. As π and e(k) commute, π corresponds one-to-one with the second part of the basis diagram πe(k). This implies that the basis diagram πe(k) is uniquely presented as a concatenation of −1 0 diagrams ω ∈ S2k+1,n and e(k). Now, the equality πe(k) = ω ω e(k) implies −1 0 0 ∗ that π = ω ω , that is, ωπ = ω . So that ωπ ∈ Bk. 4.1. An iterated inﬂation 41

∗ Lemma 4.1.4. Let σ be a permutation in Bk. Then there exists a unique 0 0 0 ∗ 0 0 0 pair (ω , π ), where ω ∈ Bk,n and π ∈ S2k+1,n, such that σ = ω π .

∗ Proof. Observe that σe(k) is a basis diagram in Vk in which `(σe(k)) = `(σ) by Lemma 1.4.8(1). Now suppose that there are two pairs (ω0, π0) and (ω, π) satisfying

σ = ω0π0 = ωπ, (4.1.2)

0 ∗ 0 0 where ω, ω ∈ Bk,n and π, π ∈ S2k+1,n. Suppose that ω 6= ω then 0 ∗ ωe(k) 6= ω e(k) by definition of Bk,n. This implies that two diagrams ωe(k) 0 and ω e(k) differ from position of horizontal edges in their top rows. By ∗ definition of diagrams in Dk,n, the position of horizontal edges in the top 0 row of the basis diagram ωe(k) (or ω e(k)) is unchanged after concatenating it with an arbitrary element of S2k+1,n on the right. This implies that 0 0 0 0 ωe(k)π differs from ω e(k)π , that is, ωπe(k) 6= ω π e(k). However, the equality 0 0 0 (4.1.2) yields ωπe(k) = ω π e(k), a contradiction. Thus ω = ω , and hence, π = π0 by multiplying (4.1.2) with ω−1 on the left.

Lemma 4.1.5. Let k, l be non-negative integers, 0 ≤ l, k ≤ [n/2]. Let ∗ ω be a permutation in Bl,n and π be a permutation in S2l+1,n. Then there 0 ∗ 0 exist ω ∈ Bk,n and π ∈ S2k+1,n such that

0 0 ωπe(k) = ω π e(k).

∗ Proof. Observe that ωπe(k) is a basis diagram in Vk by concatenation of diagrams ω, π and e(k). By Lemma 1.4.8(1) there exists a unique element ∗ σ ∈ Bk such that ωπe(k) = σe(k) and `(ωπe(k)) = `(σ). Lemma 4.1.4 implies that σ can be rewritten in the form σ = ω0π0, where ω0 and π0 are ∗ elements uniquely determined in Bk,n and S2k+1,n, respectively. Therefore, 0 0 σe(k) = ωπe(k) = ω π e(k).

∗ Note that given ω ∈ Bl,n and π ∈ S2l+1,n, Lemma 4.1.5 can be obtained via a direct calculation using the deﬁning relations as follows: By Corollary 4.1.3, ωπ is in the form

∗ ωπ = tn−1tn−2...t2lt2l−2t2l−4...t2 ∈ Bl ,

where tj = 1 or tj = sij ,j for 1 ≤ ij ≤ j ≤ n − 1. Concatenation of ωπ and 0 0 0 0 0 0 e(k) will transform ωπe(k) into σe(k), where σ = tn−1tn−2...t2kt2k−2t2k−4...t2 ∗ 0 0 is in Bk with tj = 1 or tj = sij ,j for 1 ≤ ij ≤ j ≤ n − 1. This process is done by using defining relations on the Brauer algebra Dn(N), including (S0), (S1), (S2), (3) and (5). In fact, two final relations yield the vanishing of some transpositions si in ωπe(k) and the three first imply an rearrange ωπe(k) into σe(k). 42 Cellular structure of the q-Brauer algebra

∗ Example 4.1.6. We ﬁx the element ω = s7s5,6s4,5s1,4s2 in B2,8 and the element π = s6,7s5 of S5,8. In the Brauer algebra D8(N) the diagram ωπe(2) corresponds to the diagram d0 = d∗π which is the result of concatenating d∗ and π as follows:

• • • • • •• • WWWWW OO OO ? ∗ WWWW OOO OOO ?? d WWWWW OOO OOO ? • • • •WWWW • O • O • • • • • • • •• • YYYYYY WWWWW YYYYYY WWWW 0 = YYYYY WWWWW d . • • • • • • • • • •• •••••YYYYYY WWWW ? OOo ?? oooOOO π ?oo OOO • •••••••ooo O

Using the algorithm in Section 3.3, we obtain the element ∗ 0 0 σ = s4,7s6s1,5s3,4s2 ∈ B2 satisfying d = σe(2) and `(d ) = `(σ) = 13. By direct calculation using relations (S1), (S2) on the Brauer algebra Dn(N), it also transforms ωπ into σ as follows:

(S2) ωπ = (s7s5,6s4,5s1,4s2)(s6,7s5) = s7s5,6s4,5s6,7s1,5s2

(S2) (S1) = s7s5s4(s6s5s6)s7s1,5s2 = s7s5s4(s5s6s5)s7s1,5s2

(S1) (S2) = s7(s4s5s4)s6s5s7s1,5s2 = s4s5s7s4(s6s7)s5s1,5s2

(S2) (S1) = s4s5(s7s6s7)s4s5s1,3s4s5s2 = s4s5(s6s7s6)s4s1,3(s5s4s5)s2

(S1) (S2) = s4,7s6s4s1,3(s4s5s4)s2 = s4,7s6s1,2(s4s3s4)s5s4s2

(S1) (S2) = s4,7s6s1,2(s3s4s3)s5s4s2 = s4,7s6s1,5s3,4s2.

Thus

∗ ωπ = s4,7s6s1,5s3,4s2 ∈ B2 . (4.1.3)

0 Subsequently, concatenating two diagrams d and e(3) produces the diagram d00.

• • • • • •• • YYYYYY WWWWW 0 YYYYYY WWWW d YYYYY WWWWW • • • • YY•YYYY • •WWWW • • • • • ••• • WWWW WWWWW 00 = WWWWW d . • •• •• •• • • •• •• •••WWWW e(3) • •• •• •••

0 The algorithm in Section 3.3 identifies a unique element σ = s4,7s6s3,4s2 ∗ 00 0 00 0 in B3 which satisfies d = σ e(3) and `(d ) = `(σ ) = 8. In another way, a direct calculation using the defining relations (S0), (S1), (S2), (3) and (5) 4.1. An iterated inflation 43

also yields the same result. In detail,

(S2) ωπe(3) = s4,7s6s1,5s3,4s2e(3) = s4,7s6s1,4s3s2(s5s4e(3))

(5) (S1) = s4,7s6s1,4s3s2(s3s4e(3)) = s4,7s6s1,4(s2s3s2)s4e(3)

(S2) (S1) = s4,7s6s1,3s2(s4s3s4)s2e(3) = s4,7s6s1,3s2(s3s4s3)s2e(3)

(S2) (S1), (5) = s4,7s6s1,2(s3s2s3)s4(s3s2e(3)) = s4,7s6s1,2(s2s3s2)s4(s1s2e(3))

(S0), (S2) (S1), (S2) = s4,7s6s1s3s4(s2s1s2)e(3) = s4,7s6s3s4s1(s1s2s1)e(3)

(S0) (3) (S1), (S2) = s4,7s6s3,4s2(s1e(3)) = s4,7s6s3,4s2e(3) = s7s4,6s3,4s2(s7e(3))

Thus, we obtain

00 0 ∗ 0 0 0 d = d e(3) = d πe(3) = ωπe(3) = σ e(3) = ω π e(3), (4.1.4)

0 0 where ω = s7s4,6s3,4s2 and π = s7.

Notice that computing the diagrams σ and σ0 using the algorithm in Section 3.3 can be done directly.

Lemma 4.1.7. There exists a bijection between the q-Brauer algebra

Brn(r, q) and R-vector space

[n/2] ∗ ⊕k=0 Vk,n ⊗R Vk,n ⊗R H2k+1,n(q).

∗ Proof. For a value k the dimension of each Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) is calculated by the formula:

∗ ∗ dimRVk,n ⊗R Vk,n ⊗R H2k+1,n(q) = dimR(Vk,n) · dimR(Vk,n) · dimRH2k+1,n(q) n! = ( )2 · (n − 2k)!. 2k(n − 2k)!k!

In the Brauer algebra Dn(N) the number of diagrams d which has exactly n! 2k horizontal edges is ( )2 · (n − 2k)!. Hence 2k(n − 2k)!k!

[n/2] ∗ dimR(⊕k=0 Vk,n ⊗R Vk,n ⊗R H2k+1,n(q)) = dimRDn(N) = 1 · 3 · 5...(2n − 1).

Theorem 3.8(a) in [39] implies the dimension of the Brauer algebra and the q-Brauer algebra is the same. Therefore

[n/2] ∗ dimR ⊕k=0 Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) = dimRBrn(r, q). 44 Cellular structure of the q-Brauer algebra

Now an explicit bijection will be given. Suppose that d is a diagram ∗ with a unique reduced expression (ω1, ω(d), ω2), where ω1 ∈ Bk,n, ω2 ∈ Bk,n and ω(d) ∈ S2k+1,n. As indicated in Section 3.1, the partial diagrams d1 = ω1e(k) and d2 = e(k)ω2 with `(d1) = `(ω1) and `(d2) = `(ω2) are basis ∗ diagrams of Vk,n (Vk,n), respectively. Therefore, the diagram d corresponds one-to-one to a basis element

d1 ⊗ d2 ⊗ gω(d) = ω1e(k) ⊗ e(k)ω2 ⊗ gω(d)

∗ of the R-vector space Vk,n ⊗R Vk,n ⊗R H2k+1,n(q). Now, the correspondence between an arbitrary diagram d in the Brauer algebra and a basis element gd = gω1 gω(d) e(k)gω2 of the q-Brauer algebra shown in Theorem 3.1.4, implies [n/2] ∗ a bijection from Brn(r, q) to ⊕k=0 Vk,n⊗RVk,n⊗RH2k+1,n(q) linearly spanned by the rule

gd = gω1 gω(d) e(k)gω2 7−→ ω1e(k) ⊗ e(k)ω2 ⊗ gω(d) .

From now on, if no confusion can arise, we denote by gd both a basis element of the q-Brauer algebra Bn(r, q) and its corresponding representation ∗ in Vk,n ⊗R Vk,n ⊗R H2k+1,n(q). ∗ The R-bilinear form for Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) Now we want to construct an R-bilinear form

∗ ϕk : Vk,n ⊗R Vk,n −→ H2k+1,n(q) for each 0 ≤ k ≤ [n/2]. ∗ Given elements ω1, ω2 ∈ Bk,n, by Lemma 4.1.7 we form the element ∗ ∗ ∗ Xj := dj ⊗dj ⊗1 in Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) for j = 1, 2, where dj = ωje(k) −1 and dj = e(k)ωj . The corresponding basis element of Xj in the q-Brauer algebra is Xj = gω e(k)g −1 . Then j ωj

X1X2 = (gω e(k)g −1 )(gω e(k)g −1 ). (4.1.5) 1 ω1 2 ω2 Using Lemma 2.2.3 for j = k implies X e(k)g −1 gω e(k) ∈ H2k+1,n(q)e(k) + Hn(q)e(m)Hn(q). ω1 2 m≥k+1

Hence, e(k)g −1 gω e(k) can be rewritten as an R-linear combination of the ω1 2 form X 0 e(k)g −1 gω e(k) = ajgω e(k) + a , ω1 2 (cj ) j 0 where aj ∈ R, gω(c ) ∈ H2k+1,n(q), and a is a linear combination of basis P j elements in m≥k+1 Hn(q)e(m)Hn(q). This implies that

X 0 X X1X2 = gω ( ajgω e(k))g −1 + gω a g −1 = ajgω gω e(k)g −1 + a, 1 (cj ) ω2 1 ω2 1 (cj ) ω2 j j 4.1. An iterated inﬂation 45

P where a is an R-linear combination in m≥k+1 Hn(q)e(m)Hn(q). By Deﬁnition 3.1.3, the elements gω gω e(k)g −1 , denoted by gc , are basis 1 (cj ) ω2 j elements in the q-Brauer algebra, and hence, the product X1X2 can be P rewritten to be j ajgcj + a. Using Lemma 4.1.7, gcj can be expressed in the form −1 gc = ω1e(k) ⊗ e(k)ω ⊗ gω . j 2 (cj )

Finally, X1X2 can be presented as

X −1 X1X2 = ω1e(k) ⊗ e(k)ω ⊗ ajgω + a. (4.1.6) 2 (cj ) j

∗ Subsequently, ϕk : Vk,n ⊗R Vk,n −→ H2k+1,n(q) is an R-bilinear form deﬁned by

∗ X ϕk(d1, d ) = ajgω ∈ H2k+1,n(q). (4.1.7) 2 (cj ) j

In a particular case r − 1 ϕ (e , e ) = ( )k ∈ H (q). (4.1.8) k (k) (k) q − 1 2k+1,n

Also note that, the equality (4.1.6) and Deﬁnition 4.1.7 yield the product

X1X2 in the q-Brauer algebra Brn(r, q)

∗ ∗ X1X2 = gω ϕk(d1, d )e(k)g −1 + a = gω e(k)ϕk(d1, d )g −1 + a. (4.1.9) 1 2 ω2 1 2 ω2

Deﬁnition 4.1.8. Deﬁne Jk to be an R − module generated by the basis elements gd in the q-Brauer algebra Brn(r, q), where d is a diagram whose number of vertical edges, say ϑ(d), are less than or equal n − 2k.

It is clear that Jk+1 ⊂ Jk and Jk is an ideal in Brn(r, q). By Lemma P[n/2] 2.2.4, Jk = m=k Hn(q)e(m)Hn(q).

Lemma 4.1.9. Let gc = c1 ⊗ c2 ⊗ gω(c) and gd = d1 ⊗ d2 ⊗ gω(d) , where gω(c) , gω(d) ∈ H2k+1,n(q), c1 = ω1e(k), c2 = e(k)ω2, d1 = δ1e(k), d2 = e(k)δ2 ∗ with ω1, δ1 ∈ Bk,n and ω2, δ2 ∈ Bk,n. Then

gcgd = c1 ⊗ d2 ⊗ gω(c) ϕk(c2, d1)gω(d) (mod Jk+1).

Proof. Note that the basis diagram c2 = e(k)ω2 ∈ Vk,n (similarly ∗ d1 = δ1e(k) ∈ Vk,n) can be seen as elements

gc2 = e(k) ⊗ e(k)ω2 ⊗ 1 and gd1 = δ1e(k) ⊗ e(k) ⊗ 1

∗ in Vk,n ⊗R Vk,n ⊗R H2k+1,n(q). Lemma 4.1.7 implies that gc2 = e(k)gω2 and gd1 = gδ1 e(k) are corresponding basis elements in the q-Brauer algebra 46 Cellular structure of the q-Brauer algebra

Brn(r, q), respectively. Applying Lemma 2.2.3 for j = k, the product of gc2

and gd1 is X gc2 gd1 = e(k)gω2 gδ1 e(k) ∈ H2k+1,n(q)e(k) + Hn(q)e(m)Hn(q). m≥k+1 Therefore,

(4.1.9) g −1 gc gd g −1 = (g −1 e(k)gω )(gδ e(k)g −1 ) = g −1 ϕk(c2, d1)e(k)g −1 + a, ω2 2 1 δ1 ω2 2 1 δ1 ω2 δ1

where a is a linear combination of basis elements in Jk+1 and ϕk(c2, d1) is the R-bilinear form deﬁned above. This means

0 gc2 gd1 = ϕk(c2, d1)e(k) + a (4.1.10)

0 −1 −1 with a = (g −1 ) a(g −1 ) ∈ Jk+1. As a consequence, gcgd is formed as a ω2 δ1 product of basis elements:

gcgd = (gω1 gω(c) e(k)gω2 )(gδ1 gω(d) e(k)gδ2 )

= gω1 gω(c) (e(k)gω2 gδ1 e(k))gω(d) gδ2

= gω1 gω(c) (gc2 gd1 )gω(d) gδ2 (4.1.10) 0 = gω1 gω(c) (ϕk(c2, d1)e(k) + a )gω(d) gδ2 00 = gω1 gω(c) ϕk(c2, d1)e(k)gω(d) gδ2 + a

00 with a ∈ Jk+1. Thus, by Lemma 4.1.7, gcgd can be expressed as

gcgd ≡ c1 ⊗ d2 ⊗ gω(c) ϕk(c2, d1)gω(d) (mod Jk+1).

Lemma 4.1.7 implies that Brn(r, q) has a decomposition as R-modules : ∼ [n/2] ∗ Brn(r, q) = ⊕k=0 Vk,n ⊗R Vk,n ⊗R H2k+1,n(q). ∗ Lemma 4.1.10. Let ω be an arbitrary permutation in Bk,n and π ∈ S2k+1,n. Then gωgπe(k) is a basis element in the q-Brauer algebra Brn(r, q).

Proof. Corollary 4.1.3 yields that the basis diagram d = ωπe(k) satisﬁes `(d) = `(ωπ) = `(ω) + `(π). This means the pair (ω, π) is a reduced expression of d. Therefore, by Deﬁnition 3.1.3 we get the precise statement. Lemma 4.1.11. Let k, l be non-negative integers, 0 ≤ l, k ≤ [n/2]. Let ω ∗ be a permutation in Bl,n and π a permutation in S2l+1,n. Then X g g e = a g 0 g 0 e , ω π (k) j ωj πj (k) j

0 ∗ 0 where aj ∈ R, ωj ∈ Bk,n, and πj ∈ S2k+1,n. 4.1. An iterated inﬂation 47

Proof. Corollary 4.1.3 implies `(ωπ) = `(ω) + `(π). As a consequence, gωgπ = gωπ by applying Lemma 1.3.10(1). The remainder of proof follows from the correspondence between the classical Brauer algebra Dn(N) and the q-Brauer algebra Brn(r, q) in the following way: Using the common properties of the Brauer algebra Dn(N) in Chapter 1, Section 1.4 and of the q-Brauer algebra Brn(r, q) in Chapter 2, hence, the eﬀect of the basis element gωπ on e(k) on the left(right) is similar to this of diagram (permutation) ωπ with respect to diagram e(k), respectively, on the left (right).

In fact, the operations used to move the diagram ωπe(k) into the diagram 0 0 ω π e(k) in Lemma 4.1.5 are (S0), (S1), (S2), (3) and (5) on the Brauer algebra Dn(N). In the same way, the product gωπe(k) transforms into the P 2 form a g 0 g 0 e via using corresponding relations g = (q − 1)g + q in j j ωj πj (k) i i Deﬁnition 1.3.9(iii), (H1)and (H2) on the Hecke algebra of the symmetric group as well as two relations in Lemmas 3.2.1(2) and 2.2.2(3).

Example 4.1.12. Continue considering the same diagram as in Example ∗ 4.1.6 with ω = s7s5,6s4,5s1,4s2 ∈ B2,8 and π = s6,7s5 ∈ S5,8. As in (4.1.3), the product of gω and gπ in the q-Brauer algebra Brn(r, q) is

Lem1.3.10(1) (4.1.3) + + + gωgπ = gωπ = g4,7g6g1,5g3,4g2.

Hence,

+ + + (H2) + + gωπe(3) = g4,7g6g1,5g3,4g2e(3) = g4,7g6g1,4g3g2(g5g4e(3)) Lem1.3.10(3) + + (H1) + + = g4,7g6g1,4g3g2(g3g4e(3)) = g4,7g6g1,4(g2g3g2)g4e(3) (H2) + + (H1) + = g4,7g6g1,3g2(g4g3g4)g2e(3) = g4,7g6g1,3g2(g3g4g3)g2e(3) (H2) + + (H1), Lem2.2.2(2) + + = g4,7g6g1,2(g3g2g3)g4(g3g2e(3)) = g4,7g6g1,2(g2g3g2)g4(g1g2e(3)) Def1.3.9(iii), (H2) + = g4,7g6g1((q − 1)g2 + q)g3g4(g2g1g2)e(3) (H1) + = g4,7g6g1((q − 1)g2 + q)g3g4(g1g2g1)e(3) Lem3.2.1(2) + = qg4,7g6g1((q − 1)g2 + q)g3g4g1g2e(3) + 2 + = q(q − 1)g4,7g6g1g2g3g4g1g2e(3) + q g4,7g6g1g3g4g1g2e(3) (H2) + + + 2 + + = q(q − 1)g4,7g6g1,4g1,2e(3) + q g4,7g6g3,4g1g1g2e(3) Def1.3.9(iii) + + + 2 + + = q(q − 1)g4,7g6g1,4g1,2e(3) + q g4,7g6g3,4((q − 1)g1 + q)g2e(3) + + + 2 + + + 3 + + = q(q − 1)g4,7g6g1,4g1,2e(3) + q (q − 1)g4,7g6g3,4g1,2e(3) + q g4,7g6g3,4g2e(3)

Thus, in the q-Brauer algebra Brn(r, q) the element gωπe(3) is rewritten as ∗ an R- linear combination of elements gωj e(3) (1 ≤ j ≤ 3), where ωj ∈ B3 . ∗ Now, using Lemmas 4.1.5 and 4.1.10, each element ωj of B3 can be uniquely 48 Cellular structure of the q-Brauer algebra

0 0 0 ∗ 0 expressed in the form ωj = ωjπj, where ωj ∈ B3,8 and πj ∈ S7,8, as follows:

3 X + + + g e = a g 0 g 0 e = q(q − 1)g g g g (g e ) (4.1.11) ωπ (3) j ωj πj (3) 7 4,6 1,4 1,2 7 (3) j=1 2 + + + 3 + + + q (q − 1)g7g4,6g3,4g1,2(g7e(3)) + q g7g4,6g3,4g2(g7e(3)).

N Note that if ﬁx r = q and q → 1, then Brn(r, q) ≡ Dn(N). In this case gi becomes the transposition si and the element e(k) can be identiﬁed with the diagram e(k). Hence, the last Lemma coincides with Lemma 4.1.5. This means the equality (4.1.11) in the above example recovers the equality (4.1.4) in Example 4.1.6.

The next statement shows how to get an ideal in Brn(r, q) from an ideal in Hecke algebras.

Proposition 4.1.13. Let I be an ideal in H2k+1,n(q). Then

∗ Jk+1 + Vk,n ⊗R Vk,n ⊗ I is an ideal in Brn(r, q).

Proof. Given two elements gc = c1 ⊗ c2 ⊗ gω(c) with c ∈ Dn(N) and

ϑ(c) = n − 2l, and gd = d1 ⊗ d2 ⊗ gω(d) with d ∈ Dn(N) and ϑ(d) = n − 2k, we need to prove out that:

(c1 ⊗ c2 ⊗ gω(c) )(d1 ⊗ d2 ⊗ gω(d) ) ≡ b ⊗ d2 ⊗ agω(d) (mod Jk+1)

∗ for some b ∈ Vk , and a is an element in H2k+1,n(q) which is independent of gω(d) . This property is shown via considering the multiplication of the basis elements of the q-Brauer algebra Brn((r, q) as in the proof of Lemma 4.1.7. Assume that

gc = gω1 e(l)gω(c) gω2 and gd = gδ1e(k) gω(d) gδ2 are basis elements on Brn(r, q), where ω(d) ∈ S2k+1,n; ω(c) ∈ S2l+1,n; −1 ∗ −1 ∗ ω1, ω2 ∈ Bl,n and δ1, δ2 ∈ Bk,n. Then, it implies

gcgd = gω1 (e(l)gω(c) gω2 gδ1 e(k))gω(d) gδ2 .

In the following we consider two separate cases of l and k. Case 1: If l > k, then Corollary 3.2.3(6) implies that

X e(l)gω(c) gω2 gδ1 e(k) ∈ e(l)H2k+1,n(q) + Hn(q)e(m)Hn(q), m≥l+1 4.1. An iterated inﬂation 49 and hence

gcgd = gω1 (e(l)gω(c) gω2 gδ1 e(k))gω(d) gδ2 X ∈ gω1 e(l)H2k+1,n(q)gω(d) gδ2 + Hn(q)e(m)Hn(q) m≥l+1 X ⊆ Hn(q)e(l)Hn(q) + Hn(q)e(m)Hn(q) m≥l+1 X l>k = Hn(q)e(m)Hn(q) = Jl ⊆ Jk+1. m≥l

Thus, in this case we obtain gcgd ≡ 0 (mod Jk+1). Case 2: If l ≤ k, then by Lemma 2.2.3, X e(l)gω(c) gω2 gδ1 e(k) ∈ H2l+1,ne(k) + Hn(q)e(m)Hn(q) m≥k+1 and

gcgd = gω1 (e(l)gω(c) gω2 gδ1 e(k))gω(d) gδ2 X ∈ gω1 H2l+1,ne(k)gω(d) gδ2 + Hn(q)e(m)Hn(q). m≥k+1

Without loss of generality we may assume that

X 0 X 0 gcgd = gω ( bigω e(k))gω gδ + b = bi(gω gω e(k))gω gδ + b , 1 (ci) (d) 2 1 (ci) (d) 2 i i

0 P where bi ∈ R, gω ∈ H2l+1,n, and b ∈ Hn(q)e(m)Hn(q). (ci) m≥k+1

Using Lemma 4.1.11 with respect to ω1 ∈ Bl and ω(ci) ∈ S2l+1,n, it implies that X 0 0 gω1 gω(c ) e(k) = a(i,j)g g e(k), i ω(i,j) π(i,j) j

0 ∗ 0 where a(i,j) ∈ R, ω(i,j) ∈ Bk,n, and π(i,j) ∈ S2k+1,n for l ≤ k. Finally, gcgd can be rewritten as an R - linear combination of basis elements in the q-Brauer algebra Brn(r, q) as follows:

X 0 gcgd = bi(gω gω e(k))gω gδ + b 1 (ci) (d) 2 i X X 0 0 0 = bi a(i,j)g g e(k) gω(d) gδ2 + b ω(i,j) π(i,j) i j X X 0 = b a (g 0 g 0 e g g ) + b i (i,j) ω(i,j) π(i,j) (k) ω(d) δ2 i j X 0 = b a g 0 g 0 g e g + b , i (i,j) ω(i,j) π(i,j) ω(d) (k) δ2 i, j 50 Cellular structure of the q-Brauer algebra

0 0 ∗ where π(i,j) and ω(d) are in S2k+1,n; bia(i,j) ∈ R; ω(i,j) ∈ Bk,n and δ2 ∈ Bk,n. Lemma 4.1.7 implies that the corresponding element of the product gcgd in ∗ Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) is

X 0 g g ≡ b a (ω e ⊗ e δ ⊗ g 0 g ) mod J c d i (i,j) (i,j) (k) (k) 2 π(i,j) ω(d) k+1 i, j X 0 ≡ (ω e ⊗ e δ ⊗ (b a )g 0 g ) mod J (i,j) (k) (k) 2 i (i,j) π(i,j) ω(d) k+1 i, j

≡ b ⊗ d2 ⊗ agω(d) (mod Jk+1),

P 0 ∗ P where b = ω e ∈ V , a = b a g 0 ∈ H (q), and i, j (i,j) (k) k i, j i (i,j) π(i,j) 2k+1,n d2 = e(k)δ2.

The following lemma describes the eﬀect of the involution i of the ∗ q-Brauer algebra on Vk,n ⊗R Vk,n ⊗R H2k+1,n(q).

Lemma 4.1.14. 1. If gd = d1 ⊗ d2 ⊗ gω(d) with a reduced expression −1 −1 d = (ω1, ω(d), ω2) in Dn(N). Then i(gd) = d2 ⊗ d1 ⊗ i(gω(d) ) with −1 −1 ∗ −1 −1 d2 = ω2 e(k) ∈ Vk,n and d1 = e(k)ω2 ∈ Vk,n.

2. The involution i on H2k+1,n(q) and the R −bilinear form ϕk have the following property:

−1 −1 iϕk(c, d) = ϕk(d , c )

∗ for all c ∈ Vk,n and d ∈ Vk,n.

Proof. Part (1). The basis element corresponding to the reduced expression d = (ω1, ω(d), ω2) is gd = gω1 e(k)gπ(d) gω2 . As shown in the equality (3.2.12) of Proposition 3.2.2, the image of gd via involution i is

i(gd) = g −1 e(k)g −1 g −1 ω2 ω(d) ω1

−1 −1 −1 −1 −1 −1 with `(d2 ) = `(ω2 e(k)) = `(ω2 ) and `(d1 ) = `(e(k)ω1 ) = `(ω1 ). ∗ −1 −1 ∗ Deﬁnitions of Vk,n and Vk,n imply that d1 ∈ Vk,n and d2 ∈ Vk,n. Hence, by Lemma 4.1.2 the element i(gd) can be rewritten as

−1 −1 ∗ i(gd) = d2 ⊗ d1 ⊗ i(gω(d) ) ∈ Vk,n ⊗R Vk,n ⊗R H2k+1,n(q).

Part (2). Using the same argument as in the proof of Lemma 4.1.9, c ∗ and d can be expressed as basis elements of Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) with gc = e(k) ⊗ c ⊗ 1 and gd = d ⊗ e(k) ⊗ 1, where 1 is identity in H2k+1,n(q). As a consequence,

Lem4.1.9 gcgd = (e(k) ⊗ c ⊗ 1)(d ⊗ e(k) ⊗ 1) = e(k) ⊗ e(k) ⊗ ϕk(c, d)(mod Jk+1). 4.2. Main results 51

This means the corresponding basis element in the q-Brauer algebra is

gcgd = e(k)ϕk(c, d) + a with a ∈ Jk+1. And hence,

i(gcgd) = i(e(k)ϕk(c, d) + a) = i(ϕk(c, d))i(e(k)) + i(a) P rop3.2.2 Lem2.1.10 = i(ϕk(c, d))e(k) + i(a) = e(k)i(ϕk(c, d)) + i(a).

∗ Thus, the corresponding element in Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) is of the form

i(gcgd) = i(e(k) ⊗ e(k) ⊗ ϕk(c, d)) = e(k) ⊗ e(k) ⊗ iϕk(c, d)(mod Jk+1) (4.1.12)

since i(a) ∈ Jk+1. −1 In another way, the part (1) implies that i(gc) = c ⊗ e(k) ⊗ 1 and −1 i(gd) = e(k) ⊗ d ⊗ 1. Therefore,

−1 −1 i(gd)i(gc) = (e(k) ⊗ d ⊗ 1)(c ⊗ e(k) ⊗ 1) (4.1.13) Lem4.1.9 −1 −1 = e(k) ⊗ e(k) ⊗ ϕk(d , c )(mod Jk+1).

Now the equality i(gcgd) = i(gd)i(gc) shows that

−1 −1 e(k) ⊗ e(k) ⊗ iϕk(c, d) = e(k) ⊗ e(k) ⊗ ϕk(d , c ),

−1 −1 that is, iϕk(c, d) = ϕk(d , c ). The below statement gives an 0iterated inflation0 structure for the

q-Brauer algebra Brn(r, q), and the proof of this comes from above results. Proposition 4.1.15. Let R be a commutative ring. Assume more that r, q and (r − 1)/(q − 1) are invertible in R. Then the q-Brauer algebra

Brn(r, q) over R is an iterated inﬂation of Hecke algebras of the symmetric group algebras. More precisely: as a free R-module, Brn(r, q) is equal to

∗ ∗ ∗ V0,n ⊗R V0,n ⊗R Hn(q)⊕V1,n ⊗R V1,n ⊗R H3,n(q)⊕V2,n ⊗R V2,n ⊗R H5,n(q)⊕...,

∗ and the iterated inflation starts with Hn(q), inflates it along Vk,n ⊗R Vk,n and so on, ending with an inflation of R = Hn,n(q) or R = Hn+1,n(q) as bottom layer (depending on whether n is odd or even), where H2k+1,n(q) is Hecke algebra with generators g2k+1, g2k+2... gn−1.

Corollary 4.1.16. The statement in Proposition 4.1.15 holds on Brn(N).

4.2 Main results

Before giving the main result we need the following lemma. This lemma is shown in [42] as a condition to ensure that an algebra has cellular structure. 52 Cellular structure of the q-Brauer algebra

Lemma 4.2.1. (Xi [42], Lemma 3.3) Let A be a Λ − algebra with an involution i. Suppose there is a decomposition

m A = ⊕j=1V(j) ⊗Λ V(j) ⊗Λ Bj (direct sum of Λ − modules)

where V(j) is a free Λ−modules of ﬁnite rank and Bj is a cellular Λ−algebra with respect to an involution δ and a cell chain J j ⊂ J j... ⊂ J j = B for j 1 2 sj j t each j. Deﬁne Jt = ⊕j=1V(j) ⊗Λ V(j) ⊗Λ Bj. Assume that the restriction t of i on ⊕j=1V(j) ⊗Λ V(j) ⊗Λ Bj is given by w ⊗ v ⊗ b −→ v ⊗ w ⊗ δj(b). If for each j there is a bilinear from ϕj : V(j) ⊗Λ V(j) −→ Bj such that δj(ϕj(w, v)) = ϕj(v, w) for all w, v ∈ V(j) and that the multiplication of two elements in V(j) ⊗Λ V(j) ⊗Λ Bj is governed by φj modulo Jj−1; that is, for x, y, u, v ∈ V(j), and b, c ∈ Bj, we have

(x ⊗ y ⊗ b)(u ⊗ v ⊗ c) = x ⊗ v ⊗ bϕj(y, u)c

j modulo the ideal Jj−1, and if V(j) ⊗Λ V(j) ⊗Λ Jl + Jj−1 is an ideal in A for all l and j, then A is a cellular algebra.

Theorem 4.2.2. Suppose that Λ is a commutative noetherian ring which contains R as a subring with the same identity. If q, r and (r − 1)/(q − 1) are invertible in Λ, then the q-Brauer algebra Brn(r, q) over the ring Λ is cellular with respect to the involution i.

∗ Proof. From the above construction, we know that Vk,n and Vk,n has the same ﬁnite rank. Now apply Lemma 4.2.1 to the q-Brauer algebra Brn(r, q) with j = k. Set Bn−2k = H2k+1,n(q) with 0 ≤ k ≤ [n/2], then the q-Brauer algebra Brn(r, q) has a decomposition

∗ ∗ ∗ Brn(r, q) = V0,n⊗ΛV0,n⊗ΛHn(q)⊕V1,n⊗ΛV1,n⊗ΛH3,n(q)⊕V2,n⊗ΛV2,n⊗ΛH5,n(q)⊕

∗ ∗ ...⊕V[n/2]−1,n⊗ΛV[n/2]−1,n⊗ΛH2[n/2]−1,n(q)⊕V[n/2],n⊗ΛV[n/2],n⊗ΛH2[n/2]+1,n(q), ∗ where Hn,n(q) = Hn+1,n(q) = Λ and V0,n = V0,n = Λ. Lemmas 4.1.2, 4.1.7, 4.1.9, 4.1.14, Proposition 4.1.13 and Theorem 1.3.12 show that the conditions of Lemma 4.2.1 applied for q-Brauer algebra are satisﬁed. Thus, the q-Brauer algebra Brn(r, q) is a cellular algebra.

Corollary 4.2.3. Suppose that Λ is a commutative noetherian ring which contains R as a subring with the same identity. If q, r and [N] are invertible

in Λ, then the q-Brauer algebra Brn(N) over the ring Λ is cellular with respect to the involution i.

Remark 4.2.4. By Remark 2.1.3(2), in the case q = 1 the statements in

Corollaries 4.1.16 and 4.2.3 recover these for the Brauer algebra Dn(N) due to Koenig and Xi ([26], Theorem 5.6). 4.2. Main results 53

As a consequence of Theorem 4.2.2, we have the following parametrization of cell modules for q-Brauer algebra. Given a natural number n, denote by I the set {(n − 2k, λ)| k ∈ N with 0 ≤ k ≤ [n/2], λ ` (n − 2k) with the shape (λ1, λ2, ..., λl)}.

Corollary 4.2.5. Let r, q and (r − 1)/(q − 1) be invertible elements over a commutative Noetherian ring R. The q-Brauer algebra Brn(r, q) over R has the set of cell modules

∗ {∆k(λ) = Vk,n ⊗ dk ⊗ ∆(λ)|(n − 2k, λ) ∈ I}

where dk is non-zero elements in Vk,n and ∆(λ) are cell modules of the Hecke algebra H2k+1,n(q) corresponding to the partition λ of (n − 2k).

Let us describe here what is the R-bilinear form, say Φ(k,λ), on cellular q-Brauer algebra Brn(r, q). A general deﬁnition of the R-bilinear form induced by cellular algebras is deﬁned in [15].

Deﬁnition 4.2.6. Keep the above notations. Deﬁne

Φ(k,λ) : ∆k(λ) × ∆k(λ) −→ R determined by

∗ Φ(k,λ)(dk ⊗ b ⊗ x, c ⊗ dk ⊗ y) := φ(k,λ)(xϕk(b, c), y)

∗ is an R-bilinear form, where (dk ⊗ b ⊗ x) and (c ⊗ dk ⊗ y) are in ∆k(λ) and φ(k,λ) is the symmetric R-bilinear form on the Hecke algebra H2k+1,n(q).

In the following we assume that R is a ﬁeld which contains non-zero elements q, r. Recall that e(q) to be the least positive integer m such that 2 m−1 [m]q = 1+q+q ...+q = 0 if that exists, let e(q) = ∞ otherwise. Notice that, q can be also seen as an e(q)−th primitive root of unity and e(q) = ∞

if q is not an m − th root of unity for all m. A partition λ = (λ1, λ2, ..., λl) is called e(q)-restricted if λj − λj+1 < e(q), for all j.

Theorem 4.2.7. Let Brn(r, q) be the q-Brauer algebra over an arbitrary ﬁeld R of characteristic p ≥ 0. More assume that q, r and (r − 1)/(q − 1)

are invertible in R. Then the non-isomorphic simple Brn(r, q)-modules are parametrized by the set {(n − 2k, λ) ∈ I| λ is an e(q)-restricted partition of (n - 2k)}.

Proof. By Theorem 4.2.2, the q-Brauer algebra Brn(r, q) is cellular. Then, it follows from Corollary 4.2.5 and ([15], Theorem 3.4) that the simple

Brn(r, q)-modules are parametrized by the set {(n−2k, λ) ∈ I | Φ(k,λ) 6= 0}.

If n − 2k 6= 0, then given a pair of basis elements gc = e(k) ⊗ e(k) ⊗ gω(c) and 54 Cellular structure of the q-Brauer algebra

∗ gd = e(k) ⊗ e(k) ⊗ gω(d) in Vk,n ⊗R Vk,n ⊗R H2k+1,n(q) with gω(c) , gω(d) ∈ ∆(λ), it follows from Deﬁnition 4.2.6 and Lemma 4.1.9 that

Φ(k,λ)(gc, gd) = Φ(k,λ)(e(k) ⊗ e(k) ⊗ gω(c) , e(k) ⊗ e(k) ⊗ gω(d) )

= φ(k,λ)(gω(c) ϕk(e(k), e(k)), gω(d) ) (4.1.8) r − 1 = ( )kφ (g , g ). q − 1 (k,λ) ω(c) ω(d)

The last formula implies that Φ(k,λ) 6= 0 if and only if the corresponding linear form φ(k,λ) for the cellular algebra H2k+1,n(q) is not zero. By using a result of Dipper and James ([7], Theorem 7.6) that states that φ(k,λ) 6= 0 if and only if λ is an e(q)-restricted partition of (n − 2k), it yields Φ(k,λ) 6= 0 if and only if the partition λ of n − 2k is e(q)-restricted. If n − 2k = 0 then r − 1 the last formula above implies Φ = ( )(n/2) 6= 0. Hence, we obtain (k,λ) q − 1 the precise statement.

Remark 4.2.8. The last theorem points out that all simple Brn(r, q)- modules are labeled by Young diagrams [λ] with n, n − 2, n − 4, ... boxes, where each Young diagram [λ] with e(q)-restricted partition λ of n − 2k indexes a simple module ∆(λ) of the Hecke algebra H2k+1,n(q) The next consequence follows from applying Theorem 4.2.2 and a result in [25], Proposition 3.2 on cellular algebras:

Corollary 4.2.9. Let Brn(r, q) be the q-Brauer algebra over a ﬁeld R. As- sume more that r, q, and (r − 1)/(q − 1) are invertible elements. Then the determinant of the Cartan matrix C of the q-Brauer algebra is a positive integer, where the entries of C are by deﬁnition the multiplicities of composition factors in indecomposable projective modules.

As another consequence of Theorem 4.2.2 and [15], Theorem 3.8, we obtain the following corollary.

Corollary 4.2.10. Under the assumption of the Theorem 4.2.2, the q-Brauer algebra is semisimple if and only if the cell modules are simple and pairwise non-isomorphic. Chapter 5

Quasi-heredity of the q-Brauer algebra over a ﬁeld

Let R be a field and q, r are non-zero elements in R. Sometimes, a cellular structure also provides a quasi-hereditary structure. For instance, Koenig and Xi [23, 25] and Xi [42, 43] showed that with suitable choices of parameters Brauer algebras, Partition algebras, Temperley-Lieb algebras of type A and BMW-algebras are quasi-hereditary. These results motivate to ask for a quasi-hereditary structure of the q-Brauer algebra. The question is that what conditions the q-Brauer algebra is quasi-hereditary? In this chapter we give the answer for the q-Brauer algebra Brn(r, q). We remark that this result holds true for the other versions. Let us first recall the definition of quasi-hereditary algebras introduced in [6] due to Cline, Parshall and Scott. Examples of quasi-hereditary algebras are given in [33, 6]. Definition 5.0.11. Let A be a finite dimensional R-algebra. An ideal J in A is called a heredity ideal if J is idempotent, J(rad(A))J = 0 and J is a projective left (or right) A-module. The algebra A is called quasi-hereditary provided there is a finite chain 0 = J0 ⊂ J1 ⊂ J0 ⊂ ... ⊂ Jn = A of ideals in A such that Ji+1/Ji is a hereditary ideal in A/Ji for all i. Such a chain is then called a heredity chain of the quasi-hereditary algebra A. Subsequently, we state below when the q-Brauer algebra is quasi hereditary. Recall that the Hecke algebra Hn(q) is semisimple if and only if e(q) > n (Dipper and James, [7] or [8] for more details). Theorem 5.0.12. Let R be an arbitrary field of any characteristic. As- sume moreover that q, r, and (r − 1)/(q − 1) are invertible elements in R.

Then the q-Brauer algebra Brn(r, q) is quasi-hereditary if and only if e(q) is strictly bigger than n.

Proof. Suppose that the q-Brauer algebra is quasi-hereditary. For any choice of non-zero parameter, the q-Brauer algebra has a quotient which

55 56 Quasi-heredity of the q-Brauer algebra over a ﬁeld

is isomorphic to the Hecke algebra Hn(q) of the symmetric group algebra. This quotient actually arises as Brn(r, q)/J1 for some ideal in the cell chain. Furthermore, we know by [25] that any cell chain is a hereditary chain and also note that a self-injective algebra is quasi-heredity if and only if it is semisimple. Thus, as a consequence, the Hecke algebra Hn(q) is semisimple, that is e(q) > n. Conversely, if e(q) is strictly bigger than n, then using Corollary

1.3.16(3) the Hecke algebra H2k+1,n(q) is semisimple. To prove that under our assumption the algebra Brn(r, q) is quasi-hereditary, we need to show ∗ 2 by [25] that the square (Vk,n ⊗R Vk,n ⊗R H2k+1,n(q)) is not zero modulo Jk+1 for all 0 ≤ k ≤ [n/2]. Proceed as in [25], let

λ {gS,T | λ = (λ1, λ2, ..., λl) is a partition of n − 2k, and S,T ∈ Std(λ)} be a cellular basis of the semisimple cellular algebra H2k+1,n(q). Then by λ λ λ λ Theorem 3.8 [15] there are two elements gS,T and gU,V such that gS,T gU,V µ is not zero modulo the span of all g 0 0 , where µ strictly smaller than λ S ,T 0 0 ∗ and S ,T are standard tableau of shape µ . Take the element e(k) in Vk,n λ λ (Vk,n) and consider the product of e(k) ⊗ e(k) ⊗ gS,T and e(k) ⊗ e(k) ⊗ gU,V . By Lemma 4.1.9, it implies

λ λ x := (e(k) ⊗ e(k) ⊗ gS,T )(e(k) ⊗ e(k) ⊗ gU,V ) λ λ ≡ e(k) ⊗ e(k) ⊗ gS,T ϕk(e(k), e(k))gU,V (4.1.8) r − 1 ≡ e ⊗ e ⊗ gλ ( )kgλ (k) (k) S,T q − 1 U,V r − 1 ≡ e ⊗ e ⊗ ( )k(gλ gλ )(mod J ). (k) (k) q − 1 S,T U,V k+1 r − 1 Since 6= 0, x is non-zero modulo J . q − 1 k+1 As a consequence of the last theorem and ([25], Theorem 3.1), we obtain the following corollary.

Corollary 5.0.13. Let R be any ﬁeld. Moreover, assume that q, r, and (r − 1)/(q − 1) are non-zero in R. If e(q) > n, then

1. Brn(r, q) has ﬁnite global dimension.

2. The Cartan determinant of Brn(r, q) is 1.

3. The simple Brn(r, q) − modules can be parametrized by the set of all pairs (n − 2k, λ) where 0 ≤ k ≤ [n/2] and λ is an e(q)-restricted

partition of n − 2k of the form (λ1, λ2, ..., λl).

We obtain the similar results for the version Brn(N) as follows: 57

Corollary 5.0.14. Let R be any ﬁeld. Assume more that q, r, and [N]

are non-zero in R. Then the q-Brauer algebra Brn(N) is quasi-hereditary if and only if e(q) is strictly bigger than n.

Corollary 5.0.15. Let R be any ﬁeld. Assume more that q, r, and [N] are non-zero in R. If e(q) > n, then

1. Brn(N) has ﬁnite global dimension.

2. The Cartan determinant of Brn(N) is 1.

3. The simple Brn(N) − modules can be parametrized by the set of all pairs (n − 2k, λ) with 0 ≤ k ≤ [n/2] and λ is an e(q)-restricted

partition of n − 2k of the form (λ1, λ2, ..., λl). Remark 5.0.16. Note that if q = 1 then

0 2 m−1 [m]q = 1 + 1 + 1 ... + 1 = m = 0 and hence e(q) is equal to the characteristic of R. As a consequence, the result in Corollary 5.0.14 recovers this of the Brauer algebra Dn(N) with the non-zero parameter due to Koenig and Xi ([25], Theorem 1.3). Chapter 6

A Murphy basis

This chapter is devoted to answer the following two questions about the q-Brauer algebra.

Question 6.0.17. How to give a combinatorial and direct proof for parametrization of simple modules of the q-Brauer algebra shown in Theorem 4.2.7?

Question 6.0.18. In general, does there exist an algebra isomorphism between the q-Brauer algebra and the BMW-algebra?

Theorem 4.2.7 states that all non-isomorphic simple q-Brauer modules are indexed by the e(q)-restricted partitions of n−2k where k is an integer, 0 ≤ k ≤ [n/2]. Its proof needs to use the structure ”iterated inﬂation” of the q-Brauer algebra which is complicated. Here we give a simple answer for the question 6.0.17 in Theorem 6.2.1 via using the combinatorial language which does not relate at all to the structure of the q-Brauer algebra. The question 6.0.18 arises from the similarity between the q-Brauer algebra and the BMW-algebra, such as having the same dimension, the same cellular structure, and the same index set of simple modules, up to isomorphism. To answer this second question, we will ﬁrst give a criterion for being semisimple of the q-Brauer algebra in the case n ∈ {2, 3} (Propositions 6.3.1, 6.3.2, 6.3.3). Then, Examples 6.3.5, 6.3.6, 6.3.7 yield the answer:

Claim 6.0.19. In general, there does not exist an algebra isomorphism between the q-Brauer algebra and the BMW- algebra.

To obtain the results above, we start by introducing a new basis for the q-Brauer algebra, called Murphy basis. This basis is a lift of the Mur- phy bases of the Hecke algebra of the symmetric groups ([28] or [31]). The main result stated in Theorem 6.1.10 is that: The q-Brauer algebra over a commutative ring has a basis consisting of elements that are indexed by two pairs, in each pair the ﬁrst entry is a standard tableaux

58 6.1. Main theorem 59

and the other one is a certain Brauer diagram. Since this basis exists for every version (one or two parameters) of the q-Brauer algebra, it is convenient to use the same technique for studying any version. After verifying that the Murphy basis is a cell basis, we can apply the theory of cellular algebra to produce cell (Specht) and simple modules of the q- Brauer algebra over a ﬁeld of any characteristic. This provides a combinatorial approach to study the representation theory of the q-Brauer algebra like those for the Hecke algebra of the symmetric group and its q- Schur algebra or the cyclotomic q-Schur algebra or the BMW-algebra. 2 2 For purposes of this chapter we prefer to work on the version Brn(r , q ) of the q-Brauer algebra. We freely use results obtained in the previous 2 2 chapters for the version Brn(r , q ) without further comments or proofs. Throughout, let R be an arbitrary ﬁeld of any characteristic p ≥ 0.

6.1 Main theorem

Recall that if k is an integer, 0 ≤ k ≤ [n/2], then Jk is an R-module generated by the basis elements gd of the q-Brauer algebra, where d is a Brauer diagram whose number of vertical edges are less than or equal n−2k

(Deﬁnition 4.1.8). Then Jk is an ideal of the q-Brauer algebra on version 2 2 Brn(r , q ) and has the form

[n/2] X 2 2 Jk = Hn(q )e(j)Hn(q ). (6.1.1) j=k

2 2 The following statements on the q-Brauer algebra Brn(r , q ) are necessary for references in this section.

Lemma 6.1.1. The following statements hold for the q-Brauer algebra 2 2 Brn(r , q ).

2 −1 −1 −2 1. g2j+1e(k) = e(k)g2j+1 = q e(k), and g2j+1e(k) = e(k)g2j+1 = q e(k) for 0 ≤ j < k;

r − r−1 2. e e = e e = ( )je for any j ≤ k; (j) (k) (k) (j) q − q−1 (k)

+ + − − 3. g2i−1,2je(k) = g2j+1,2ie(k) and g2i−1,2je(k) = g2j+1,2ie(k) for 1 ≤ i ≤ j < k;

+ + − − 4. e(k)g2l,1 = e(k)g2,2l+1 and e(k)g2l,1 = e(k)g2,2l+1 for l < k;

−1 −1 −1 −1 5. e(k)g2jg2j−1 = e(k)g2jg2j+1 and e(k)g2j g2j−1 = e(k)g2j g2j+1 for 1 ≤ j < k; 60 A Murphy basis

+ + − − 6. e(k)g2j,2i−1 = e(k)g2i,2j+1 and e(k)g2j,2i−1 = e(k)g2i,2j+1 for 1 ≤ i ≤ j < k;

r − r−1 7. e g− g+ e = ( )j−1e for 1 ≤ j < k; (k) 2k,2j−1 2k+1,2j (j) q − q−1 (k+1)

r − r−1 8. e g e = rq( )j−1e for 1 ≤ j ≤ k; (k) 2j (j) q − q−1 (k)

2 2 P 2 2 9. e(k)Hn(q )e(j) ⊂ e(k)H2j+1,n(q ) + m≥k+1 Hn(q )e(m)Hn(q ), where j ≤ k;

− + 10. e(k+1) = e(k)g2k,1g2k+1,2e.

An equivalent statement of Lemma 4.1.11 is the following.

Lemma 6.1.2. Let k, l be integers, 0 < k ≤ l ≤ [n/2], and let u be

a permutation in Bl,n and π a permutation in S2l+1,n. Then there exist a(ω,u) ∈ R, for v ∈ Bk,n and ω ∈ S2k+1,n, such that X e(k)gπgu = a(ω,v)e(k)gωgv.

ω∈S2k+1,n v∈Bk,n

2 2 Lemma 6.1.3. Let k be an integer, 0 < k ≤ [n/2]. If b ∈ Brn(r , q ), u ∈ Bk,n, then there exist a(ω,v) ∈ R, for ω ∈ S2k+1,n and v ∈ Bk,n, such that X e(k)gub ≡ a(ω,v)e(k)gωgv mod Jk+1.

ω∈S2k+1,n v∈Bk,n

Proof. The proof of this lemma is a special case of the proof of Proposi-

tion 4.1.13. In detail, we choose gc := e(k)gu and gd := b

r − r−1 Theorem 6.1.4. Let R be a ﬁeld of any characteristic. If q, r and q − q−1 2 2 are invertible in R. Then the q-Brauer algebra Brn(r , q ) over R is cellular with respect to the involution i deﬁned in Proposition 3.2.2.

∗ Remark 6.1.5. By deﬁnition of the set Bk,n and Bk,n (see the formula −1 ∗ (3.3.2), (3.3.3)), it is clear that if u ∈ Bk,n then i(gu) = gu−1 and u ∈ Bk,n. ∗ From now on, we write gu := i(gu) with u ∈ Bk,n replacing element gu−1 −1 ∗ with u ∈ Bk,n. The following statement gives an explicit cellular basis for the 2 2 q-Brauer algebras Brn(r , q ). Its proof follows from Theorem 3.1.4, Propositions 3.2.2 and 4.1.13. 6.1. Main theorem 61

2 2 Theorem 6.1.6. The q-Brauer algebra Brn(r , q ) is freely generated as an R-module by the basis

∗ { gue(k)gπgv | u, v ∈ Bk,n and π ∈ S2k+1,n for 0 ≤ k ≤ [n/2] }. Moreover, the following statements hold. 1. The involution i satisﬁes

∗ ∗ i : gugπe(k)gv 7→ gv gπ−1 e(k)gu

for all u, v ∈ Bk,n and π ∈ S2k+1,n.

2 2 2. Suppose that b ∈ Brn(r , q ) and let k be an integer, 0 ≤ k ≤ [n/2]. If u, v ∈ Bk,n and π ∈ S2k+1,n, then there exist v1 ∈ Bk,n and π1 ∈ S2k+1,n such that

∗ X ∗ gue(k)gπgvb ≡ a(π1,v1)gue(k)gπ1 gv1 mod Jk+1. (6.1.2)

π1∈S2k+1,n v1∈Bk,n

We state some notations which are necessary for later work. Deﬁnition 6.1.7. Let k be an integer, 0 ≤ k ≤ [n/2].

1. Recall that Jk+1 is an R-module with a basis

∗ {gue(l)gπgv | u, v ∈ Bl,n and π ∈ S2l+1,n for all k < l ≤ [n/2]}. (6.1.3)

2. Let Λn := {(k, λ) | for 0 ≤ k ≤ [n/2], and λ a partition of n − 2k}. For (k, λ), (l, µ) ∈ Λn deﬁne an ordered relation in Λn by

(k, λ) ¤ (l, µ) if λ ¤ µ,

where λ ¤ µ was determined in Deﬁnition 1.3.7. We will write (k, λ) £ (l, µ) to mean that λ £ µ. by Deﬁnition 1.3.7

it is clear that the order relation ¤ in Λn is a partial order. We call ¤ the dominance order for the q-Brauer algebra.

3. For (k, µ) ∈ Λn, deﬁne the element

mµ = e(k)cµ where cµ is deﬁned in (1.3.3). (6.1.4)

4. For (k, λ) ∈ Λn, deﬁne In(k, λ) to be the set of ordered pairs

In(k, λ) = Std(λ) × Bk,n = {(s, u): s ∈ Std(λ) and u ∈ Bk,n} . (6.1.5) 62 A Murphy basis

Example 6.1.8. Let n = 10 and µ = (3, 2, 1). The example in (1.3.2) yields the subgroup Sµ = hs5, s6, s8i, and X mµ = e(2) gσ = e(2)(1 + g5)(1 + g6 + g6g5)(1 + g8).

σ∈Sµ ˇ λ Let Brn be an R-module with spanning set µ ∗ ∗ (s, u), (t, v) ∈ In(l, µ) x(s,u)(t,v) := gugd(s)mµgd(t)gv . (6.1.6) µ £ λ for (l, µ), (k, λ) ∈ Λn ˇ λ ˇ λ Lemma 6.1.9. Suppose that (k, λ) ∈ Λn, then Jk+1 ⊆ Brn and Brn is an 2 2 ideal of the q-Brauer algebra Brn(r , q ). ∗ Proof. By (6.1.3) every basis element in Jk+1 is of the form gue(l)gπgv where u, v ∈ Bl,n and π ∈ S2l+1,n, k+1 ≤ l ≤ [n/2]. By the definition in Theorem P ∗ 1.3.14, the element gπ can be rewritten as gπ = (s,t,µ) gd(s)cµgd(t) with (l, µ) ∈ Λn. Since n − 2l < n − 2k, Definition 1.3.7 of the dominance order implies that µ £ λ. Thus, ∗ P ∗ ∗ P µ ˇ λ gue(l)gπgv = (s,t,µ) gugd(s)mµgd(t)gv = (s,t,µ) x(s,u)(t,v) ∈ Brn, ˇ λ meaning Jk+1 ⊆ Brn. To prove the second statement, it is sufficient to show that the product µ ˇ λ µ ˇ λ x(s,u)(t,v) ·b is in Brn, where x(s,u)(t,v) ∈ Brn (with µ£λ) and b is an arbitrary 2 2 basis element of the q-Brauer algebra Brn(r , q ). Lemma 6.1.3 implies

X (e(l)gv)b ≡ a(π1,v1)e(l)gπ1 gv1 mod Jl+1.

π1∈S2l+1,n v1∈Bl,n

Notice that in the formula (1.3.5) of Theorem 1.3.14, we have c1t = 1cµgd(t). We have the calculation:

µ ∗ ∗ ∗ ∗ x(s,u)(t,v) · b = (gugd(s)mµgd(t)gv)b = (gugd(s)cµgd(t))(e(l)gvb) X ∗ ∗ = a(π1,v1)gugd(s)e(l)(1cµgd(t)gπ1 )gv1 + Jl+1

π1∈S2l+1,n v1∈Bl,n

(1.3.5) X ∗ ∗ = a(π1,v1)gugd(s)e(l)(c1tgπ1 )gv1 + Jl+1

π1∈S2l+1,n v1∈Bl,n (1.3.6) X ∗ ∗ X ˇµ = a(π1,v1)gugd(s)e(l)( at1 c1t1 + H2l+1,n)gv1 + Jl+1 π1∈S2l+1,n t1∈Std(µ) v1∈Bl,n (1.3.5) X ∗ ∗ X ˇµ = a(π1,v1)gugd(s)e(l)( at1 cµgd(t1) + H2l+1,n)gv1 + Jl+1 π1∈S2l+1,n t1∈Std(µ) v1∈Bl,n 6.1. Main theorem 63

X X ∗ ∗ = at1 a(π1,v1)(gugd(s)e(l)cµgd(t1)gv1 )

π1∈S2l+1,n t1∈Std(µ) v1∈Bl,n X ∗ ∗ ˇµ + a(π1,v1)(gugd(s)e(l)H2l+1,ngv1 ) + Jl+1 π1∈S2l+1,n v1∈Bl,n

(6.1.4),(1.3.4) X X ∗ ∗ = at1 a(π1,v1)(gugd(s)mµgd(t1)gv1 )

π1∈S2l+1,n t1∈Std(µ) v1∈Bl,n X X ∗ ∗ + a(π1,v1)a(s2,t2)(gugd(s2)mµ2 gd(t2)gv1 ) + Jl+1 π1∈S2l+1,n µ2`n−2l, µ2£µ v1∈Bl,n s2,t2∈Std(µ2) X X = a a (xµ ) t1 (π1,v1) (s,u)(t1,v1) π1∈S2l+1,n t1∈Std(µ) v1∈Bl,n X X + a a (xµ2 ) + J , (π1,v1) (s2,t2) (s2,u)(t2,v1) l+1 π1∈S2l+1,n µ2`n−2l, µ2£µ v1∈Bl,n s2,t2∈Std(µ2)

ˇµ 2 where a(π1,v1), at1 and as2,t2 are in R, and H2l+1,n is the ideal of H2l+1,n(q ) deﬁned in (1.3.4). By Lemma 6.1.9 and the assumption µ £ λ (hence ˇ λ l ≥ k), all elements occuring in the last formula are in Brn, and hence, so µ is x(s,u)(t,v) · b. The main result of this section is the following.

2 2 Theorem 6.1.10. The q-Brauer algebra Brn(r , q ) is freely generated as an R–module by the collection λ ∗ ∗ x = g g mλgd(t)gv (s, u), (t, v) ∈ In(k, λ), for (k, λ) ∈ Λn . (s,u)(t,v) u d(s) (6.1.7)

Moreover, the following statements hold.

λ λ λ 1. The involution i sends x(s,u)(t,v) to i(x(s,u)(t,v)) = x(t,v)(s,u) for all (t, v), (s, u) ∈ In(k, λ).

2 2 2. Suppose that b ∈ Brn(r , q ), (k, λ) ∈ Λn and (s, u), (t, v) ∈ In(k, λ).

Then there exist a(t2,v2) ∈ R, (t2, v2) ∈ In(k, λ) such that

λ X λ ˇ λ x(s,u)(t,v) · b ≡ a(t2,v2)x(s,u)(t2,v2) mod Brn. (6.1.8) (t2,v2)∈In(k,λ) 64 A Murphy basis

2 2 Proof. Let b be an arbitrary element in the q-Brauer algebra Brn(r , q ). Then by Theorem 6.1.6, b can be expressed as an R-linear combination of basis elements g∗ g e g , uj πj (kj ) vj X b = a g∗ g e g , (6.1.9) j uj πj (kj ) vj j

where (kj, λ) ∈ Λn, aj ∈ R, πj ∈ S2k+1,n, and uj, vj ∈ Bkj ,n. Using the deﬁnition in Theorem 1.3.14, the element gπj has an expression

g = g∗ c g πj d(sj ) λj d(tj ) with sj, tj ∈ Std(λj). Replace gπj in (6.1.9) by the last formula, X X b = a g∗ g∗ c g e g = a g∗ g∗ e c g g j uj d(sj ) λj d(tj ) (kj ) vj j uj d(sj ) (kj ) λj d(tj ) vj j j (6.1.4) X X = a g∗ g∗ m g g = a xλ , j uj d(sj ) λj d(tj ) vj j (sj ,uj )(tj ,vj ) j j where (sj, uj), (tj, vj) ∈ In(kj, λj). Therefore, the set (6.1.7) linearly spans the q-Brauer algebra 2 2 Brn(r , q ). The independence of (6.1.7) follows from the linear indepen- dences in Theorems 1.3.14 and 6.1.6. The statement (1) is obtained by combining the involution i on the 2 Hecke algebra Hn(q ) and Lemma 6.1.1(10). The statement (2) is shown as follows: Let (k, λ), (l, µ) ∈ Λn. As the same arguments in the proof above, it suffices to consider the product xλ · b, where b = xµ with (s , u ), (t , v ) ∈ I (l, µ) is a basis (s,u)(t,v) (s1,u1)(t1,v1) 1 1 1 1 n 2 2 element in Brn(r , q ). Subsequently, consider two cases with respect to partitions µ and λ. The first case is µ B λ: Then Definition 1.3.7 of the dominance order implies that k ≤ l. So, by Lemma 2.2.3 we get

∗ 2 X e(k)gvgu1 e(l) ∈ H2k+1,n(q )e(l) + Hne(m)Hn. m≥l+1 Hence,

∗ ∗ 2 ∗ X (cλgd(t))(e(k)gvgu1 e(l))(gd(s1)cµ) ∈ cλgd(t)H2k+1,n(q )e(l)gd(s1)cµ + Hne(m)Hn m≥l+1 (6.1.4) 2 ∗ X ⊆ H2k+1,n(q )gd(s1)mµ + Hne(m)Hn m≥l+1 k≤l 2 ∗ X ⊆ H2k+1,n(q )gd(s1)mµ + Hne(m)Hn m≥k+1 (6.1.1) 2 ∗ ˇ λ ⊆ H2k+1,n(q )gd(s1)mµ + Jk+1 ⊆ Brn. 6.1. Main theorem 65

(6.1.4) Since (c g )(e g g∗ e )(g∗ c ) = (m g g )(g∗ g∗ m ), the last λ d(t) (k) v u1 (l) d(s1) µ λ d(t) v u1 d(s1) µ inclusion yields

(xλ )(xµ ) = (g∗g∗ m g g )(g∗ g∗ m g g ) (s,u)(t,v) (s1,u1)(t1,v1) u d(s) λ d(t) v u1 d(s1) µ d(t1) v1 ∗ ∗ ˇ λ ˇ λ ∈ gugd(s)Brngd(t1)gv1 ⊆ Brn.

λ Thus, xλ )(xµ ) ≡ 0 (mod Brˇ ), namely, (s,u)(t,v) (s1,u1)(t1,v1) n

λ ˇ λ x(s,u)(t,v) · b ≡ 0 (mod Brn).

The second case is λ D µ, that is l ≤ k: Using Lemma 6.1.1(9), it yields

∗ X e(k)gvgu1 e(l) ∈ e(k)H2l+1,n + Hne(m)Hn. m≥k+1

Applying Lemma 6.1.1(9) and the same arguments as in the previous case implies that

∗ ∗ ∗ ∗ (cλgd(t))(e(k)gvgu1 e(l))(gd(s1)cµ) = (mλgd(t)gv)(gu1 gd(s1)mµ) ∗ X ∈ cλgd(t)e(k)H2l+1,ngd(s1)cµ + Hne(m)Hn m≥k+1 (6.1.1),(6.1.4) ˇ λ ⊆ mλH2l+1,n + Jk+1 ⊆ mλH2l+1,n + Brn (by Lemma 6.1.9).

Hence,

λ ∗ ∗ ∗ ∗ x(s,u)(t,v) · b = (gugd(s)mλgd(t)gv)(gu1 gd(s1)mµgd(t1)gv1 ) ∗ ∗ ˇ λ ∈ gugd(s)mλH2l+1,ngd(t1)gv1 + Brn ∗ ∗ ˇ λ ⊆ gugd(s)mλH2l+1,ngv1 + Brn.

λ It implies that x(s,u)(t,v) · b can be rewritten as an R-linear combination

λ ∗ ∗ X ˇ λ x(s,u)(t,v) · b = gugd(s)mλ aπ1 gπ1 gv1 + Brn

π1∈S2l+1,n (6.1.4) ∗ ∗ X ˇ λ = gugd(s)cλ aπ1 e(k)gπ1 gv1 + Brn

π1∈S2l+1,n Lem6.1.2 X X λ = g∗g∗ c a ( a e g g ) + Brˇ u d(s) λ π1 (ωπ1 ,vπ1 ) (k) ωπ1 vπ1 n

π1∈S2l+1,n ωπ1 ∈S2k+1,n vπ1 ∈Bk,n X X λ = a ( a g∗g∗ e (c g )g ) + Brˇ π1 (ωπ1 ,vπ1 ) u d(s) (k) λ ωπ1 vπ1 n

π1∈S2l+1,n ωπ1 ∈S2k+1,n vπ1 ∈Bk,n 66 A Murphy basis

(1.3.6) X X ∗ ∗ X ˇλ ˇ λ = aπ a(ω ,v )g g e(k)( at c1t + H )gv + Br 1 π1 π1 u d(s) ωπ1 ωπ1 2k+1,n π1 n π1∈S ωπ ∈S t ∈Std(λ) 2l+1,n 1 2k+1,n ωπ1 vπ1 ∈Bk,n X X X ∗ ∗ = aπ a(ω ,v )( at g g e(k)c1t gv ) 1 π1 π1 ωπ1 u d(s) ωπ1 π1 π1∈S ωπ ∈S t ∈Std(λ) 2l+1,n 1 2k+1,n ωπ1 vπ1 ∈Bk,n X X λ + a a g∗g∗ e ˇλ g + Brˇ π1 (ωπ1 ,vπ1 ) u d(s) (k)H2k+1,n vπ1 n

π1∈S2l+1,n ωπ1 ∈S2k+1,n vπ1 ∈Bk,n X X X λ = aπ1 a(ωπ ,vπ )( atω x(s,u)(t ,v )) 1 1 π1 ωπ1 π1 π1∈S ωπ ∈S t ∈Std(λ) 2l+1,n 1 2k+1,n ωπ1 vπ1 ∈Bk,n X X λ + a a g∗g∗ e ˇλ g + Brˇ , π1 (ωπ1 ,vπ1 ) u d(s) (k)H2k+1,n vπ1 n

π1∈S2l+1,n ωπ1 ∈S2k+1,n vπ1 ∈Bk,n

ˇ λ where aπ , a(ω ,v ), and at are in R. By the deﬁnition of Br in (6.1.6) 1 π1 π1 ωπ1 n ˇ λ it is obviously that the middle term in the last formula is in Brn. So, the last formula can be rearranged such that

λ X λ ˇ λ x(s,u)(t,v) · b ≡ a(t2,v2)x(s,u)(t2,v2) mod Brn, (t2,v2)∈In(k,λ) where t := t ∈ Std(λ), v := v ∈ B , and a is the corresponding 2 ωπ1 2 π1 k,n (t2,v2) coeﬃcient of xλ . (s,u)(t2,v2) Thus, we get the precise statement (6.1.8).

ˇ λ As a consequence of the above theorem, Brn is the R-module freely generated by the collection (6.1.6). The new basis in (6.1.7) of the q-Brauer algebra can be verified to be a cellular basis in the sense of Graham and Lehrer [15] by checking conditions of the definition of cellular algebra (see Definition 1.1.1) as follows: 2 2 The q-Brauer algebra Brn(r , q ) has the cell datum (Λn, In, C, i) where

(C1) Λn is a partially ordered set with the dominance order defined in Definition 1.3.7. For each (k, λ) ∈ Λn, In(k, λ) is a finite set satisfying that a C : In(k, λ) × In(k, λ) → Brn(N)

(k,λ)∈Λn

λ determined by the rule C((s, u), (t, v)) = x(s,u)(t,v) is injective map.

(C2) This condition follows from Theorem 6.1.10(1).

(C3) This condition is satisﬁed by Theorem 6.1.10(2). 6.1. Main theorem 67

(C3’) This condition is obtained by applying i to the equation (6.1.8), we obtain

λ X λ ˇ λ i(b) · x(t,v)(s,u) ≡ a(t2,v2)x(t2,v2)(s,u) mod Brn. (t2,v2)∈In(k,λ)

Now we can use the representation theory of cellular algebras for the 2 2 q-Brauer algebra Brn(r , q ).

2 2 Deﬁnition 6.1.11. For (k, λ) ∈ Λn deﬁne the right Brn(r , q )-module C(k, λ) as follows: C(k, λ) is a free R-module with basis n o λ ˇ λ x(t,v) := mλgd(t)gv + Brn | (t, v) ∈ In(k, λ) (6.1.10)

2 2 and with the right Brn(r , q )-action deﬁned by

λ ˇ λ X λ ˇ λ 2 2 x(t,v)b + Brn = a(t1,v1)x(t1,v1) + Brn for b ∈ Brn(r , q ), (t1,v1)∈In(k,λ)

where the coeﬃcients a(t1,v1) ∈ R, for (t1, v1) in In(k, λ), are determined by the expression (6.1.8). C(k, λ) is called a Specht (or Cell) module of the q-Brauer algebra. Theorem 6.1.10 is an analogue to that for the Hecke algebra of the 2 symmetric group Hn(q ). So, we call the set (6.1.7) a Murphy basis of the q-Brauer algebra, and call the set (6.1.13) a Murphy basis of the Specht module C(k, λ). Note that, we do not know the other properties of a Murphy basis for the q-Brauer algebra so far. Remark 6.1.12. 1. Note that the notion of Specht module C(k, λ) of the q-Brauer algebra, which is introduced in this section, is compatible with the Specht module Sλ of the Hecke algebra of the symmetric group in (1.3.7). If q = 1, then it recovers the notion of to Specht module of the classical Brauer algebra used in [18]. 2. In the case q = 1 the version of Theorem 6.1.10 for the q-Brauer 2 algebra Brn(N ) coincides with Enyang’s result on the classical Brauer algebra Dn(N) ([12], Theorem 9.1). It implies that over a ﬁeld R of any 2 characteristic the other results for the q-Brauer algebra Brn(N ) in this dissertation recover those for the classical Brauer algebra Dn(N). The example below illustrates a basis for Specht module.

Example 6.1.13. Let n = 5, k = 1, and λ = (2, 1). If j, ij are integers

with 1 ≤ ij ≤ j ≤ n − 1, write tj = 1 or tj = sjsj−1 ··· sij , so that

B2,5 = {v = t2t4| tj = 1 or tj = sj,ij , 1 ≤ ij ≤ j for j ∈ {1, 2, 4}};

B1,5 = {v = t2t3t4| tj = 1 or tj = sj,ij , 1 ≤ ij ≤ j ≤ 4}

= {1, s2, s2,3, s2,1, s2,1s3, s2,1s3,2, s2,4, s2,1s3,4, s2,1s3,2s4, s2,1s3,2s4,3}. 68 A Murphy basis

Since the set of partitions {µ |µ £ λ} = {µ1 = (3), µ2 = (1)} we obtain as follows:

With µ1 = (3) the Young subgroup Sµ1 = {1, s3, s4, s3s4, s4s3, s4s3s4}, µ1 Std(µ1) = { t = 3 4 5 }, and hence

mµ1 = e(1 + g3 + g4 + g3g4 + g4g3 + g3g4g3) = e(1 + g3)(1 + g4 + g4g3).

µ2 With µ2 = (1) the Young subgroup Sµ2 = {1}, Std(µ2) = { t = 5 }, ˇ (2,1) and mµ2 = e(2). So by (6.1.9) the two-sided ideal Br5 has a basis:

xµ1 , xµ1 , (s1,u1)(t1,v1) (t1, v1), (s1, u1) ∈ In(l, µ1), (1,u1)(1,v1) v1, u1 ∈ B1,5 µ = µ . x 2 (t , v ), (s , u ) ∈ I (l, µ ) x 2 v , u ∈ B (s2,u2)(t2,v2) 2 2 2 2 n 2 (1,u2)(1,v2) 2 2 2,5 (6.1.11) n o λ 3 4 λ 3 5 In the other hand, Std(λ) = t = 5 , t s4 = 4 and mλ = e(1 + g3), the basis for C(1, λ), of the form displayed in (6.1.13), is

ˇ (2,1) ˇ (2,1) e(1 + g3)gv + Br5 , e(1 + g3)g4gv + Br5 |v ∈ B1,5 . A key to understanding the structure of C(k, λ) is the R-bilinear form

h , iλ which we deﬁne below.

Deﬁnition 6.1.14. For (k, λ) ∈ Λn, deﬁne h , iλ : C(k, λ) × C(k, λ) → R by

λ λ λ λ ˇ λ hx(t,v), x(s,u)iλmλ ≡ x(t,v) · i(x(s,u)) mod Brn. (6.1.12) This means

ˇ λ ˇ λ ∗ ∗ ˇ λ hmλgd(t)gv + Brn, mλgd(s)gu + Brniλmλ ≡ mλgd(t)gvgugd(t)mλ + Brnmλ ∗ ∗ ˇ λ ≡ mλgd(t)gvgugd(t)mλ mod Brn.

Lemma 6.1.15. Suppose that (k, λ) ∈ Λn and x, y ∈ C(k, λ). Then

1. hx, yiλ = hy, xiλ;

∗ 2 2 2. hxb, yiλ = hx, yb iλ for all b ∈ Brn(r , q );

λ λ λ 3. xx(s,u)(t,v) = hx, x(s,u)iλx(t,v) for all (s, u), (t, v) ∈ In(k, λ). Proof. This is a special case of a general result on cellular algebras (see [15], Proposition 2.4 or [28], Proposition 2.9 for detailed proof). Example 6.1.16. Let n = 3 and λ = (1). This example calculates the Gram matrix of the Specht module C(1, (1)), using the R-bilinear form in Deﬁnition 6.1.14. (1) We obtain mλ = e, Std((1)) = {t = 3 } (hence gd(t(1)) = 1), and ˇ λ Brn = (0). Formula (3.3.3) implies that the ordered set 6.1. Main theorem 69

B1,3 = {v1 = 1, v2 = s2, v3 = s2s1}. By Deﬁnition 6.1.11 the corresponding basis of the Specht module C(1, (1)) is

n (1) (1) o x (1) := e · 1 · gv | (t , vj) ∈ I3(1, (1)) = {e, eg2, eg2g1} (6.1.13) (t ,vj ) j

(1) (1) The calculations for h x (1) x (1) iλ are as follows, using Lemma 6.1.15: (t ,vi) (t ,vj )

00 −1 (1) (1) P rop3.2.2 2 (E1) r − r h x (1) x (1) i(1)e = e · i(e) = e = e; (t ,v1) (t ,v1) q − q−1

(1) (1) (1) (1) h x (1) x (1) i(1)e = h x (1) x (1) i(1)e (t ,v1) (t ,v2) (t ,v2) (t ,v1) 00 P rop3.2.2 (E2) = e · i(eg2) = eg2e = (rq)e;

(1) (1) (1) (1) h x (1) x (1) i(1)e = h x (1) x (1) i(1)e (t ,v1) (t ,v3) (t ,v3) (t ,v1) 00 P rop3.2.2 (E2) 3 = e · i(eg2g1) = eg1g2e = (rq )e;

(1) (1) P rop3.2.2 2 h x (1) x (1) i(1)e = eg2 · i(eg2) = e(g2) e (t ,v2) (t ,v2) Def1.3.9(iii) 2 2 2 = e(q − 1)g2e + q e −1 (E )00,(E )00 r − r 1 =2 (q2 − 1)rqe + q2 e; q − q−1

(1) (1) (1) (1) h x (1) x (1) i(1)e = h x (1) x (1) i(1)e = eg2 · i(eg2g1) (t ,v2) (t ,v3) (t ,v3) (t ,v2) 00 P rop3.2.2 ((H1) (E2) 5 = eg2g1g2e = eg1g2g1e = (rq )e;

(1) (1) P rop3.2.2 2 h x (1) x (1) i(1)e = eg2g1 · i(eg2g1) = eg2(g1) g2e (t ,v3) (t ,v3) Def1.3.9(iii) 2 2 2 = eg2(q − 1)g1g2e + q e(g2) e −1 (E )00,(E )00 r − r 1 =2 (q4 − 1)rq3e + q4 e; q − q−1

(1) (1) Hence the Gram matrix h x (1) x (1) iλ of the Specht module (t ,vi) (t ,vj ) C(1, (1)) with respect to the ordered basis is  a rq rq3  r − r−1 rq q2a + (q2 − 1)rq rq5 , where a = .   q − q−1 rq3 rq5 q4a + (q4 − 1)rq3 The determinant of the Gram matrix given above is 3q5(r2 − q2)2(q4r2 − 1) (6.1.14) r3(q2 − 1)3 70 A Murphy basis

6.2 Representation theory over a ﬁeld

Using a Murphy basis of Specht modules C(k, λ), we have deﬁned the new

R-bilinear form, h , iλ, for the q-Brauer algebra. This bilinear form differs from the one given in Definition 4.2.6. In detail, instead of determining the bilinear form via the known bilinear forms, including two bilinear forms of both the Hecke algebra and the iterated inflation’s algebra, h , iλ is directly defined using a Murphy basis of Specht modules. This enables us to give explicit calculations in the proof of Theorem 6.2.1. Using the general theory of cellular algebras we obtain some results about the representation theory of q-Brauer algebras.

Denote rad(C(k, λ)) = {x ∈ C(k, λ)| hx, yiλ = 0 for all y ∈ C(k, λ)} and D(k, λ) = C(k, λ)/rad(C(k, λ)). The following are special cases of results in [15]. −1 −1 Statement 1. For (k, λ) ∈ Λn, r, q and (r − r )/(q − q ) invertible 2 2 elements in an arbitrary ﬁeld R, let Brn(r , q ) be the q-Brauer algebra over R. Then

2 2 1. rad(C(k, λ)) is a Brn(r , q )-submodule of C(k, λ); 2. If D(k, λ) 6= 0 then

(a) D(k, λ) is simple; 2 2 (b) rad(C(k, λ)) is the radical of the Brn(r , q )-module C(k, λ).

2 2 Statement 2. For (k, λ), (l, µ) ∈ Λn, let Brn(r , q ) be a q-Brauer 2 2 algebra over an arbitrary ﬁeld R. Suppose M is a Brn(r , q )-submodule of C(k, λ) and ϕ : C(l, µ) −→ C(k, λ)/M

2 2 is a Brn(r , q )-module homomorphism, and h, iµ 6= 0. Then 1. ϕ 6= 0 only if λ ¤ µ.

2. If λ = µ, then there are elements r0 6= 0 and r1 in R such that for all x ∈ C(l, µ), we have r0ϕ(x) = r1x + M.

For (k, λ), (l, µ) ∈ Λn and D(l, µ) 6= 0, let dλµ = [C(k, λ): D(l, µ)] be the composition multiplicity of D(l, µ) in C(k, λ). The next statement gives a classiﬁcation of the simple modules, up to isomorphism, of the q-Brauer algebra. This result is an analogue to that for the Hecke algebra due to Dipper and James (see [7], Theorem 7.6).

−1 −1 Theorem 6.2.1. For (k, λ) ∈ Λn, r, q and (r − r )/(q − q ) invertible 2 2 elements in an arbitrary ﬁeld R, let Brn(r , q ) be a q-Brauer algebra over R. Then 6.2. Representation theory over a ﬁeld 71

2 1. The set {D(l, µ)| (l, µ) ∈ Λn, µ is an e(q )-restricted partition} is a 2 2 complete set of pairwise non-isomorphic simple Brn(r , q )-modules.

2 2. For (k, λ), (l, µ) ∈ Λn, suppose that µ is an e(q )-restricted partition. Then dµµ = 1 and dλµ 6= 0 only if λ ¤ µ.

Proof. (1). Since the q-Brauer algebra is a cellular algebra, it follows from Theorem 3.4 in [15] that the set {D(l, µ)| D(l, µ) 6= 0 for partition µ of n − 2l, 0 ≤ l ≤ [n/2]} 2 2 is a complete set of pairwise non-isomorphic simple Brn(r , q )-modules. The remainder of proof is to show that D(l, µ) 6= 0 if and only if µ is e(q2)-restricted partition of n − 2l. µ ˇ µ Indeed, pick two non-zero elements x(s,u) = mµgd(s)gu + Brn and µ ˇ µ x(t,v) = mµgtgv + Brn in C(l, µ) with arbitrary pairs (s, u), (t, v) ∈ In(l, µ). This yields, using (6.1.12), µ µ ˇ µ ˇ µ hx(s,u), x(t,v)iµmµ = hmµgd(s)gu + Brn, mµgd(t)gv + Brniµmµ (6.2.1) (6.1.12) ∗ ∗ ˇ µ ≡ mµgd(s)gvgugd(t)mµ mod Brn (6.1.4) ∗ ∗ ˇ µ ≡ e(l)(cµgd(s))gvgu(gd(t)cµ)e(l) mod Brn (1.3.5) ∗ ˇ µ ≡ e(l)(c1sgv)(guct1)e(l) mod Brn (1.3.6) X ˇµ X ˇµ ˇ µ ≡ e(l)( as1 c1s1 + H2l+1,n)( at1 ct11 + H2l+1,n)e(l) mod Brn s1∈Std(µ) t1∈Std(µ) (1.3.7) X X ∗ ˇ µ ≡ e(l)( as1 cs1 )( at1 ct1 )e(l) mod Brn s1∈Std(µ) t1∈Std(µ) X ∗ ˇ µ ≡ e(l) as1 at1 (cs1 ct1 )e(l) mod Brn s1,t1∈Std(µ) (1.3.9) X ˇµ ˇ µ ≡ e(l) as1 at1 (hcs1 , ct1 icµ + H2l+1,n)e(l) mod Brn s1,t1∈Std(µ) −1 (6.1.4),Lem6.1.1(2) X r − r l µ ≡ a a hc , c im mod Brˇ q − q−1 s1 t1 s1 t1 µ n s1,t1∈Std(µ) where as1 , at1 are coefficients in R. Then, using Theorem 1.3.15(1) if µ is an e(q2)-restricted partition of n − 2l then Dµ 6= 0. The definition of Dµ implies that there exist µ ˇ µ s, t ∈ Std(µ) such that hcs, cti= 6 0. Now fixing x(s,1) = mµgd(s) +Brn and µ ˇ µ x(t,1) = mµgd(t) + Brn the previous calculation yields

−1 r − r µ hxµ , xµ i m ≡ lhc , c im mod Brˇ . (s,1) (t,1) µ µ q − q−1 s t µ n 72 A Murphy basis

r − r−1 It implies hx, yi = lhc , c i= 6 0, that is D(l, µ) 6= 0. µ q − q−1 s t Conversely, if µ is not e(q2)-restricted then by Theorem 1.3.15(1) the R- µ 2 bilinear form h , i on the Specht module S of the Hecke algebra H2k+1,n(q ) µ is zero. This means that hcs1 , ct1 i = 0 for any cs1 , ct1 ∈ S . By calculation µ µ µ µ in (6.2.1) it implies hx(s,u), x(t,v)iµ = 0 for all x(s,u), x(t,v) ∈ C(l, µ), namely, D(l, µ) = 0. (2). The second statement follows by applying the general theory of cellular algebra and Proposition 3.6 [15].

−1 −1 Corollary 6.2.2. For (k, λ) ∈ Λn, r, q and (r − r )/(q − q ) invertible 2 2 elements in an arbitrary ﬁeld R, let Brn(r , q ) be a q-Brauer algebra over R. The following statements are equivalent.

2 2 1. Brn(r , q ) is semisimple;

2. C(k, λ) = D(k, λ) for all (k, λ) ∈ Λn;

3. The R-bilinear form h , iλ (cf. (6.1.12)) is non-degenerate for all (k, λ) ∈ Λn.

Remark 6.2.3. The same results as Theorem 6.2.1 and Corollary 6.2.2 2 hold true for the version Brn(N ) of the q-Brauer algebra. Furthermore, when q = 1 then the result in Theorem 6.2.1 recovers that for the classical

Brauer algebra Dn(N) with non-zero parameter which was shown in [15], Theorem 4.17 by Graham and Lehrer. Also notice that in this case the Specht module of the classical Brauer algebra in [15] is dual to the one in this chapter.

6.3 Is the q-Brauer algebra generically isomorphic with the BMW-algebra?

This section starts by proving a semisimplicity criterion for the q-Brauer algebra in the case n = {2, 3}. The statements are the following.

2 2 Proposition 6.3.1. Let Brn(r , q ) be the q-Brauer algebra over R with r − r−1 invertible elements r, q and ∈ R. Then q − q−1

2 2 2 1. Br2(r , q ) is semisimple if and only if e(q ) > 2.

2 2 2 2. Br3(r , q ) is semisimple if and only if e(q ) > 3 and

3q5(r2 − q2)2(q4r2 − 1) 6= 0. r3(q2 − 1)3 6.3. Is the q-Brauer algebra generically isomorphic with the BMW-algebra? 73 Proof. If n = 2 and λ is a partition of 2, then the cell modules C(0, λ) λ coincide with the cell modules S of the Hecke algebra H2, and it hence h , i ≡ h , iλ. Therefore, by Corollary 1.3.16, the R-bilinear form h , i is 2 ˇ λ ˇλ non-degenerate if and only if e(q ) > 2. If λ = ∅, then Br2 ≡ H2 = R and mλ = e. As shown in (6.1.13), the cell module C(1, λ) has a basis

ˇ λ { egv + Br2 | v ∈ B1,2 = {1}} = {e}.

The Gram determinant with respect to this basis is

−1 −1 (E )00 r − r r − r he , ei e = e2 =1 e, that is, he , ei = . λ q − q−1 λ q − q−1 With n = 3 and λ a partition of 3, using the same argument as above yields 2 that the R-bilinear form h , iλ is non-degenerate if and only if e(q ) > 3. Otherwise, if n = 3 and λ = (1), then applying Example (6.1.16) the Gram determinant on C(1, λ) is non-zero if and only if

3q5(r2 − q2)2(q4r2 − 1) 6= 0. r3(q2 − 1)3

Hence, we get the statement (2) by using Corollary 6.2.2.

2 2 2 When replacing the version Brn(r , q ) by Brn(N ) or Brn(r, q), then the statements are the following.

Proposition 6.3.2. Let Brn(r, q) be the q-Brauer algebra over R with r − 1 invertible elements r, q and ∈ R. Then q − 1

1. Br2(r, q) is semisimple if and only if e(q) > 2.

2. Br3(r, q) is semisimple if and only if e(q) > 3 and 3q(r − q)2(q2r − 1) 6= 0. (q − 1)3

The proof is similar to the one above.

2 Proposition 6.3.3. Let N ∈ Z\{0}. Let Brn(N ) be the q-Brauer algebra over R with 0 6= q, [N 2] ∈ R. Then

1. Br2(N) is semisimple if and only if e(q) > 2.

2. Br3(N) is semisimple if and only if e(q) > 3 and

3q4(qN − q[N])([N] + qN+1 + qN+3) 6= 0.

The proof uses the same arguments as in Proposition 6.3.1. 74 A Murphy basis

Remark 6.3.4. 1. Notice that if q = 1 then e(q2) (resp. e(q)) is equal to the characteristic p of the ﬁeld R. It implies that for r = qN with N ∈ Z \{0} and the limit q → 1, our results above recover Theorems 1.4.3(1) and 1.4.4(1) for the classical Brauer algebra Dn(N) due to Rui [35] r − r−1 in the case n ∈ {2, 3}. In particular, when Lim = N and q→1 q − q−1 3q5(r2 − q2)2(q4r2 − 1) 3q5(q2N − q2)2(q4q2N − 1) Lim = Lim q→1 r3(q2 − 1)3 q→1 q3N (q2 − 1)3 3q9 (q2(N−1) − 1)2 (q2(N+2) − 1) = Lim · · = 3(N − 1)(N + 2), q→1 q3N (q2 − 1)2 (q2 − 1)

2N 2 then Brn(q , q ) ≡ Dn(N) over the ﬁeld R in which the limit q → 1 can be formed. Applying Proposition 6.3.1 it implies the following:

2N 2 • Over the complex ﬁeld Br2(q , q ) is semisimple if and only if N 6= 0 2N 2 and Br3(q , q ) is semisimple if and only if N 6∈ {−2, 0, 1}. It is straightforward to check that the results above coincide with those of Theorem 1.4.3(1) in the case n ∈ {2, 3}.

• Over arbitrary field of characteristic p > 0 which the limit q → 1 can be formed then Proposition 6.3.1 yields: 2N 2 Br2(q , q ) is semisimple if and only if N 6= 0 and p > 2. 2N 2 Br3(q , q ) is semisimple if and only if N 6∈ {−2, 0, 1} and p > 3. These results recover Theorem 1.4.4(1) for n ∈ {2, 3}. 2 In the case q = 1, Brn(N ) ≡ Dn(N) over field R of characteristic p ≥ 0. Then a direct calculation yields that Proposition 6.3.3 recovers Theorems 1.4.3(1) and 1.4.4 for n ∈ {2, 3} again. 2. Over the field of characteristic zero all results above agree with Wenzl’s results in the case n ∈ {2, 3} (see Theorem 2.2.5). The following table provides a detailed comparison. Semisimplicity Theorem 2.2.5 due to Wenzl Proposition 6.3.2

Br2(r, q) If q 6= 0, e(q) > 2 and If and only if q 6= 0, r∈ / {q−2, q−1, 0, 1, q, q2} e(q) > 2 and r∈ / {0, 1}

Br3(r, q) If q 6= 0, e(q) > 3 and If and only if q 6= 0 r∈ / {q−3, q−2, q−1, 0, 1, q, q2, q3} e(q) > 3 and r∈ / {q−2, 0, 1, q}

2 2 For the version Brn(r , q ) Wenzl’s result is the same as that for 2 2 Brn(r, q) with replacing the pair of parameters (r, q) by (r , q ) and e(q) by e(q2). 3. Propositions 6.3.1 and 6.3.2 imply a negative answer for the question about the existence of an isomorphism between the q - Brauer algebra 2 2 Brn(r , q ) (resp. Brn(r, q)) and the BMW- algebra Bn. Three following examples illustrate Claim 6.0.19. 6.3. Is the q-Brauer algebra generically isomorphic with the BMW-algebra? 75

In the two following examples, the BMW-algebra is not simple, but the q-Brauer algebra with the same parameter value is semisimple

2 2 Example 6.3.5. We consider both algebras Br3(r , q ) and B3 over the complex ﬁeld. These algebras simultaneously depend on two parameters r and q. Fixing r = q−1 and q2 = −i, then by Theorem 5.9(b) [34] the BMW-algebra B3 is not semisimple since q4 + 1 = (−i)2 + 1 = 0.

2 On the other hand, both [m]q2 = 1 + q = 1 − i 6= 0 and

2 2 2 2 2 [m]q2 = 1 + q + (q ) = 1 − i + (−i) = −i 6= 0, namely, e(q ) = m > 3.

Moreover, a direct calculation yields 3q5(r2 − q2)2(q4r2 − 1) 3q5(q−2 − q2)2(q4q−2 − 1) = = 6i 6= 0. r3(q2 − 1)3 q−3(q2 − 1)3

2 2 Therefore, applying Proposition 6.3.1(2) the q-Brauer algebra Br3(r , q ) is semisimple. The result is illustrated in the following table:

2 2 C B3 Br3(r , q ) (r, q2) = (q−1, −i) not semisimple semisimple

Example 6.3.6. With respect to the version√ Br3(r, q) and B3 over the −1 complex ﬁeld, we choose r = q and q = i i, then by Theorem√ 5.9(b) [34] 4 4 the BMW-algebra B3 is not semisimple since√ q + 1 = (i i) + 1 = 0. In other words, both [m]q = 1 + q = 1 + i i 6= 0 and √ √ √ 2 2 [m]q = 1 + q + q = 1 + i i + (i i) = i i 6= 0, namely, e(q) = m > 3.

By direct calculation, it yields 3q(r − q)2(q2r − 1) 3q(q−1 − q)2(q2q−1 − 1) √ = = 3q−1 = 3(i i)−1 6= 0. (q − 1)3 (q − 1)3

Hence, by Proposition 6.3.2(2) the q-Brauer algebra Br3(r, q) is semisimple. The next table below gives the result.

Br (r, q) C √ B3 3 (r, q) = (q−1, i i) not semisimple semisimple

Example 6.3.7. This example shows that over the ﬁeld of characteristic p = 5 the BMW-algebra is not semisimple for twelve parameter values, but the q-Brauer algebra is not semisimiple for less than four parameter values. 76 A Murphy basis

¯ ¯ Over the prime field F5 if q ∈ {2, 3}, then it is obvious that 2 2 [m]q2 = 1¯ + q = 0¯ and hence e(q ) ≤ 2. Applying Theorem 5.9 in [35] ¯ the BMW-algebra B2 is not semisimple for all r ∈ F5 \{0}. Otherwise, ¯ ¯ ¯ 2 with q ∈ F5 \{0, 2, 3} a direct calculation implies that e(q ) > 2, and by −1 Theorem 5.9 [35] B2 is not semisimple for r ∈ {q , −q} = {1¯, 4¯}. Thus, there totally exist twelve value pairs (r, q) such that the BMW-algebra B2 is not semisimple over the field F5. 2 2 By Proposition 6.3.1(1) the q-Brauer algebra Br2(r , q ) over the field ¯ ¯ ¯ F5 is not semisimple if and only if q ∈ {2, 3} and r ∈ F5 \{0} such that −1 −1 2 2 (r − r )/(q − q ) 6= 0. Direct calculation yields Br2(r , q ) over the field ¯ ¯ ¯ ¯ F5 is not semisimple for all parameters q ∈ {2, 3} and r ∈ {2, 3}. This means that there are totally such four value pairs (r, q).

Similarly, on the version Brn(r, q) Proposition 6.3.2(1) implies that the q-Brauer algebra Br2(r, q) over the ﬁeld F5 is not semisimple if and only if ¯ ¯ q ∈ {4} and r ∈ F5 \{0} such that (r−1)/(q−1) 6= 0. That is, Br2(r, q) over ¯ ¯ ¯ ¯ the ﬁeld F5 is not semisimple for all parameters q = 4 and r ∈ {2, 3, 4}. The total parameter values, such that the algebras are not semisimple, are summarized in the following table.

The non-semisimple case F5 × F5 The BMW-algebra B2 (r, q) ∈ ({1¯, 2¯, 3¯, 4¯} × {2¯, 3¯}) ∪ ({2¯, 3¯} × {1¯, 4)¯ } 2 2 The q-Brauer algebra Br2(r , q ) (r, q) ∈ {2¯, 3¯} × {2¯, 3¯}

The q-Brauer algebra Br2(r, q) (r, q) ∈ {2¯, 3¯, 4¯} × {4¯} Chapter 7

The cellularly stratiﬁed structure of the q-Brauer algebra

We consider the version Brn(N) of q-Brauer algebra. Throughout, let R be an arbitrary ﬁeld of any characteristic. Assume moreover that q and [N] are invertible in R.

7.1 The main result

An iterated inﬂation In this section we are going to construct another iterated inﬂation for

the q-Brauer algebra Brn(N). This new iterated inflation enables us to assert that the q-Brauer algebra is cellularly stratified. The main reason for this construction is that: In [19], the cellularly stratified structure just appears on ’diagram algebras’. These algebras have a basis which can be represented by certain diagrams. In particular, diagram algebras in [19] which are shown to be cellularly stratified include Brauer algebras, BMW- algebras and Partition algebras. In their proofs the conditions (E) and (I)

in Definition 1.2.1 are easily verified since multiplying two diagrams e(l) and e(k) is essentially concatenation. In the case of the q-Brauer algebra, we do not know if the q- Brauer algebra is a diagram algebra. That is, it is not known if there exists a basis represented by certain diagrams for the q-Brauer algebra. So, the iterated inflation for the q-Brauer algebra in Chapter 4, defined by certain diagrams, faces some technical difficulties in proving the conditions (E) and (I). In construction of a new iterated inflation of the q-Brauer algebra we define a general multiplication for elements on any two inflations (Definition 7.1.4). This multiplication

enables us to give a simple calculation for product of elementse ˆ(k) and eˆ(l) in the proof of the main theorem (Theorem 7.1.8).

77 78 The cellularly stratiﬁed structure of the q-Brauer algebra

1 For k an integer, 0 ≤ k ≤ [n/2], denotee ¯ := e . Notice that (k) [N]k (k) e(k) is an element of the q-Brauer algebra Brn(N) in Deﬁnition 2.1.9. Deﬁne Uk to be an R-vector space spanned by the set 1 { g | d is a basis diagram in V } [N]k d k,n

∗ 1 −1 ∗ and let U := R{ g −1 | d is a basis diagram in V }. k [N]k d k,n

Lemma 7.1.1. The R-module Uk has a basis {gωe¯(k) | ω ∈ Bk,n}. Dually, ∗ −1 the R-module Uk has a basis {e¯(k)gω | ω ∈ Bk,n}. Moreover, n! dim U = dim U ∗ = . R k R k 2k(n − 2k)!k!

Proof. Observe that each basis diagram of Vk,n is a diagram of the Brauer algebra. For a basis diagram d ∈ Vk,n, by Lemma 4.1.1 and Deﬁnition 3.1.3 gd is a basis element of the q-Brauer algebra and is of the form gd = gωe(k), where ω ∈ Bk,n. Theorem 3.1.4 implies that the elements 1 1 g (= g e ) with ω ∈ B are independent. The proof is the [N]k d [N]k ω (k) k,n ∗ same for Uk . Thus, we get the precise statement.

Proposition 7.1.2. Fix an index k and let B := Jk/Jk+1 be the R-algebra (possibly without identity). Then B is isomorphic (as R-algebra) to an ∗ inﬂation Uk ⊗R Uk ⊗R H2k+1,n(q) of the Hecke algebra of symmetric group ∗ H2k+1,n(q) along free R-modules Uk, Uk . The R-bilinear form ∗ φ : Uk ⊗R Uk −→ H2k+1,n(q) is determined by

φ(¯e(k)gv ⊗ gue¯(k))¯e(k) :=e ¯(k)gv · gue¯(k) mod Jk+1,

−1 where u , v are in Bk,n. ∼ Proof. Obviously, by (6.1.1), it implies that B = Jk/Jk+1 = Hne(k)Hn as R-vector spaces. Hence, dimRB = dimRHne(k)Hn. Using Theorem 3.8(b) in [39], dimRHne(k)Hn is equal to the number of all Brauer diagrams d which has exactly 2k horizontal edges. By direct calculation we obtain (n!)2 dim H e H = dim B = dim U ∗⊗ U ⊗ H (q) = . R n (k) n R R k R k R 2k+1,n 4k(n − 2k)!(k!)2 As a consequence, by Theorem 3.1.4 B is a vector space with R-basis

{gd : = gw1 e(k)gωgw2 | d a diagram with exactly 2k horizontal edges in Dn(N)}. 7.1. The main result 79

∗ Denote fk : B −→ Uk ⊗R Uk ⊗R H2k+1,n(q) to be a map determined by

k gw1 e(k)gωgw2 7→ [N] (gw1 e¯(k) ⊗ e¯(k)gw2 ⊗ gω),

where (w1, ω, w2) is a reduced expression of d. This definition provides an isomorphism of R-modules. In order to show fk is an algebra isomorphism, we need to define a multiplication on ∗ Uk ⊗R Uk ⊗R H2k+1,n(q) using the R-bilinear form φ as follows: −1 −1 For u1 , u2 , v1, v2 ∈ Bk,n and ω1, ω2 ∈ Σ2k+1,n, define

(gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 )·(gu2 e¯(k) ⊗ e¯(k)gv2 ⊗ gω2 ) (7.1.1)

:= (gu1 e¯(k) ⊗ e¯(k)gv2 ⊗ (gω1 φ(¯e(k)gv1 , gu2 e¯(k))gω2 ).

∗ By Theorem 3.1 in [24] the multiplication makes Uk ⊗RUk ⊗RH2k+1,n(q) into an associative algebra (possibly without identity). It is left to verify that

fk is a ring isomorphism. To this end, pick two arbitrary basis elements

gh1 , gh2 ∈ B up. Assume that gh1 = gu1 e(k)gω1 gv1 , gh2 = gu2 e(k)gω2 gv2 . Then, we obtain

gh1 · gh2 = (gu1 e(k)gω1 gv1 )(gu2 e(k)gω2 gv2 ) (7.1.2) L2.2.1(4) = (gu1 e(k)gω1 )(¯e(k)gv1 gu2 e¯(k))(e(k)gω2 gv2 ) φ = (gu1 e(k)gω1 )(φ(¯e(k)gv1 ⊗ gu2 e¯(k))¯e(k))(e(k)gω2 gv2 ) L2.2.1(4) = (gu1 e(k)gω1 )(φ(¯e(k)gv1 ⊗ gu2 e¯(k)))(e(k)gω2 gv2 ).

= (gu1 e(k))(gω1 φ(¯e(k)gv1 ⊗ gu2 e¯(k))gω2 )(e(k)gv2 ).

Since gω1 φ(¯e(k)gv1 ⊗ gu2 e¯(k))gω2 ∈ H2k+1,n(q), it can be represented as an R-linear combination of basis elements gw with w ∈ S2k+1,n. That is,

X gω1 φ(¯e(k)gv1 ⊗ gu2 e¯(k))gω2 = awgw, (7.1.3)

w∈Σ2k+1,n

where aw are coeﬃcients in R. Putting this formula into the equation (7.1.2), it yields

(7.1.3) X gh1 · gh2 = (gu1 e(k))( awgw)(e(k)gv2 ) (7.1.4)

w∈Σ2k+1,n X = aw(gu1 e(k)gwe(k)gv2 ).

w∈Σ2k+1,n 80 The cellularly stratiﬁed structure of the q-Brauer algebra

Hence,

(7.1.4) X fk(gh1 · gh2 ) = fk( aw(gu1 e(k)gwe(k)gv2 ))

w∈Σ2k+1,n X = awfk(gu1 e(k)gwe(k)gv2 )

w∈Σ2k+1,n

Lem2.2.1(4) X k = [N] awfk(gu1 e(k)gwgv2 )

w∈Σ2k+1,n

fk X 2k = [N] aw(gu1 e¯(k) ⊗ e¯(k)gv2 ⊗ gw).

w∈Σ2k+1,n In other words, we also have

fk k k fk(gh1 ) · fk(gh2 ) = [N] (gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 ) · [N] (gu2 e¯(k) ⊗ e¯(k)gv2 ⊗ gω2 )

(7.1.1) 2k = [N] gu1 e¯(k) ⊗ e¯(k)gv2 ⊗ gω1 φ(¯e(k)gv1 , gu2 e¯(k))gω2

(7.1.3) 2k X = [N] (gu1 e¯(k) ⊗ e¯(k)gv2 ⊗ awgw)

w∈Σ2k+1,n X 2k = [N] aw(gu1 e¯(k) ⊗ e¯(k)gv2 ⊗ gw).

w∈Σ2k+1,n

Hence, fk(gh1 · gh2 ) = fk(gh1 ) · fk(gh2 ). Thus, fk is an algebra isomorphism.

The next statement can be veriﬁed directly.

Lemma 7.1.3. Under fk the involution i : B → B corresponds to the ∗ involution on Uk ⊗R Uk ⊗R H2k+1,n(q) which sends gue¯(k) ⊗ e¯(k)gv ⊗ gω to −1 gv−1 e¯(k) ⊗ e¯(k)gu−1 ⊗ gω−1 , where u , v ∈ Bk,n and ω ∈ Σ2k+1,n. Recall that i is the involution determined in Proposition 3.2.2 by −1 i(e(k)) = e(k) and i(gω) = gω with ω ∈ Sn. We next show that the layers ﬁt together (which is more than just having a ﬁltration by two-sided ideals).

Deﬁnition 7.1.4. For k, l non-negative integers, 0 ≤ k, l ≤ [n/2], let ∗ ∗ x ∈ Uk ⊗R Uk ⊗R H2k+1,n(q) and y ∈ Ul ⊗R Ul ⊗R H2l+1,n(q). Deﬁne

−1 −1 x · y := fk(fk (x) · fl (y) mod Jk+1), where fj, 0 ≤ j ≤ [n/2], is the isomorphism defined in Proposition 7.1.2. Note that the product in the right hand-side of definition is the usual multiplication in the q-Brauer algebra. When k = l, the definition above recovers Definition (7.1.1). This claim is a consequence of the following lemma. 7.1. The main result 81

Lemma 7.1.5. For k, l non-negative integers, 0 ≤ k, l ≤ [n/2], let gh1 := gu1 e(k)gω1 gv1 ∈ Jk \ Jk+1 and gh2 := gu2 e(l)gω2 gv2 ∈ Jl \ Jl+1 be two basis elements in the q-Brauer algebra and let their respective f-images be gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 and gu2 e¯(l) ⊗ e¯(l)gv2 ⊗ gω2 . Then the product gh1 · gh2 either is an element of Jk+1 or is an element of Jk \ Jk+1, and in the latter case it corresponds under f to a scalar multiple of an element gu1 e¯(k) ⊗ b ⊗ gω1 c where b is an element in Uk and c is an element in H2k+1,n(q)

Proof. We separately consider two cases of l and k. Case 1. If l > k, then Lemma 2.2.3 implies that X e(k)gv1 · gu2 gω2 e(l) ∈ H2k+1,n(q)e(l) + Hne(m)Hn, m≥l+1 and hence

(gu1 e(k)gω1 gv1 ) · (gu2 e(l)gω2 gv2 ) = gu1 gω1 (e(k)gv1 · gu2 gω2 e(l))gv2 X ∈ gu1 gω1 H2k+1,n(q)e(l)gv2 + Hne(m)Hn m≥l+1 X ⊆ Hne(l)Hn + Hne(m)Hn m≥l+1 X (6.1.1) l>k = Hne(m)Hn = Jl ⊆ Jk+1. m≥l

Thus, gh1 · gh2 ≡ 0 (mod Jk+1), that is, gh1 · gh2 ∈ Jk. Case 2. If l ≤ k, then by Corollary 3.2.3(6), we obtain

X e(k)gv1 · gu2 e(l) ∈ e(k)H2l+1,n + Hne(m)Hn. m≥k+1

Hence, X e(k)gv1 · gu2 e(l)gω2 gv2 ∈ e(k)H2l+1,ngv2 + Hne(m)Hn. m≥k+1

Since v2 ∈ Bl,n, applying Lemma 6.1.2 the product e(k)gv1 · gu2 gω2 e(l)gv2 can be rewritten as an R-linear combination of elements of the form e(k)gω3 gv3 where v3 ∈ Bk,n and ω3 ∈ Σ2k+1,n. This means X e(k)gv1 gu2 gω2 e(l)gv2 = a(ω3,v3)e(k)gω3 gv3 + a, (7.1.5)

ω3∈Σ2k+1,n v3∈Bk,n

where a(ω3,v3) are coeﬃcients in R and a is an R-linear combination in Jk+1. 82 The cellularly stratiﬁed structure of the q-Brauer algebra

Now, the product of gh1 and gh2 is computed as follows.

gh1 · gh2 = (gu1 e(k)gω1 gv1 )·(gu2 e(l)gω2 gv2 ) (7.1.6)

= gu1 gω1 (e(k)gv1 · gu2 gω2 e(l)gv2 ) (7.1.5) X = gu1 gω1 a(ω3,v3)e(k)gω3 gv3 + a

ω3∈Σ2k+1,n v3∈Bk,n X = a(ω3,v3)gu1 gω1 e(k)gω3 gv3 + gu1 gω1 a,

ω3∈Σ2k+1,n v3∈Bk,n

with gu1 gω1 a ∈ Jk+1. By Deﬁnition 7.1.4 we obtain

(gu1 e¯(k)⊗e¯(k)gv1 ⊗ gω1 ) · (gu2 e¯(l) ⊗ e¯(l)gv2 ⊗ gω2 ) −1 −1 = fk fk (gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 ) · fl (gu2 e¯(l) ⊗ e¯(l)gv2 ⊗ gω2 ) mod Jk+1

= fk(gh1 · gh2 mod Jk+1) (7.1.6) X = fk a(ω3,v3)gu1 gω1 gω3 e(k)gv3

ω3∈Σ2k+1,n v3∈Bk,n

fk X k = [N] (gu1 e¯(k) ⊗ e¯(k)gv3 ⊗ a(ω3,v3)gω1 gω3 )

ω3∈Σ2k+1,n v3∈Bk,n

= gu1 e¯(k) ⊗ b ⊗ gω1 c,

P P k where b := ω3∈Σ2k+1,n e¯(k)gv3 and c := ω3∈Σ2k+1,n [N] a(ω3,v3)gω3 . v3∈Bk,n v3∈Bk,n Corollary 7.1.6. If k = l then Deﬁnition 7.1.4 recovers Deﬁnition (7.1.1).

Proof. Keep notations as in the last lemma. For k = l we get

(gu1 e¯(k)⊗e¯(k)gv1 ⊗ gω1 ) · (gu2 e¯(k) ⊗ e¯(k)gv2 ⊗ gω2 ) (7.1.7) −1 −1 = fk fk (gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 ) · fk (gu2 e¯(l) ⊗ e¯(k)gv2 ⊗ gω2 ) mod Jk+1

= fk(gh1 · gh2 mod Jk+1).

For k = l, the deﬁnition of the bilinear form φ yields

φ 2k φ(¯e(k)gv1 ⊗ gu2 e¯(k))¯e(k) = [N] e(k)gv1 gu2 e(k) mod Jk+1. (7.1.8)

Hence,

gh1 · gh2 mod Jk+1 ≡ (gu1 e(k)gω1 gv1 ) · (gu2 e(k))gω2 gv2 mod Jk+1 (7.1.9)

≡ gu1 gω1 (e(k)gv1 · gu2 e(k)gω2 gv2 ) modJk+1

(7.1.8) −2k = gu1 gω1 ([N] φ(¯e(k)gv1 ⊗ gu2 e¯(k))¯e(k))gω2 gv2 . 7.1. The main result 83

Now substituting (7.1.9) into Formula (7.1.7) yields

(gu1 e¯(k) ⊗ e¯(k)gv1 ⊗ gω1 ) · (gu2 e¯(k) ⊗ e¯(k)gv2 ⊗ gω2 ) = −2k = fk gu1 gω1 ([N] φ(¯e(k)gv1 ⊗ gu2 e¯(k))¯e(k))gω2 gv2

fk −k = [N] (gu1 e¯(k) ⊗ e¯(k)gv2 ) ⊗ gω1 φ(¯e(k)gv1 ⊗ gu2 e¯(k))gω2 .

Altogether we have proved the following theorem.

Theorem 7.1.7. The q-Brauer algebra Brn(N) is an iterated inﬂation of the Hecke algebras of symmetric groups. More precisely: as a free

R- module, Brn(N) is equal to

∗ ∗ Hn(q) ⊕ (U1 ⊗R U1 ⊗R H3,n(q)) ⊕ (U2 ⊗R U2 ⊗R H5,n(q)) ⊕ ...,

∗ and the iterated inﬂation starts with Hn(q), inﬂates it along U1 ⊗R U1 ⊗R

H3,n(q) and so on, and ends with an inﬂation of R = Hn+1,n(q) or R = Hn,n(q) as bottom layer, depending on whether n is even or odd.

Now, the main result in this chapter is the following.

Theorem 7.1.8. Let R be an arbitrary ﬁeld of characteristic p ≥ 0. Let q

and [N] be invertible in R. Then the q-Brauer algebra Brn(N) over R is cellularly stratiﬁed.

Proof. The proof proceeds by verifying Deﬁnition 1.2.1 of a cellularly stratiﬁed algebra. The assumption (C) follows from Theorem 7.1.7. To

show that the q-Brauer algebra Brn(N) satisﬁes the condition (E), for n, k integers, 1 ≤ k ≤ [n/2] denotee ˆ(k) :=e ¯(k) ⊗ e¯(k) ⊗ 1. It is obvious that

2 (ˆe(k)) = (¯e(k) ⊗ e¯(k) ⊗ 1) · (¯e(k) ⊗ e¯(k) ⊗ 1) (7.1.1) =e ¯(k) ⊗ e¯(k) ⊗ 1φ(¯e(k) ⊗ e¯(k))1

=e ¯(k) ⊗ e¯(k) ⊗ 1 =e ˆ(k)

2 where φ(¯e(k) ⊗e¯(k))¯e(k) = (¯e(k)) =e ¯(k). This implies the condition (E). The condition (I) is straightforward to verify as follows: By Lemma 2.2.1(4) a calculation yields that, for k ≥ l

−k −l −(k+l) e¯(k)e¯(l) = ([N] e(k))([N] e(l)) = [N] e(k)e(l) (7.1.10)

2.2.1(4) −(k+l) −k = [N] e(l)e(k) = [N] e(k) =e ¯(k).

By Proposition 7.1.2 the elementse ˆ(k) ande ˆ(l) have corresponding pre- −k −k images [N] e(k) and [N] e(l) respectively. So, using the same arguments 84 The cellularly stratiﬁed structure of the q-Brauer algebra

as in the proof of Lemma 7.1.5 the product of idempotentse ˆ(k) ande ˆ(l) is computed as follows. For k ≥ l,

eˆ(k) · eˆ(l) = (¯e(k) ⊗ e¯(k) ⊗ 1) · (¯e(l) ⊗ e¯(l) ⊗ 1) Def7.1.4 −1 −1 = fk fk (¯e(k) ⊗ e¯(k) ⊗ 1) · fl (¯e(l) ⊗ e¯(l) ⊗ 1) mod Jk+1 f −1 k −k −l = fk([N] e(k) · [N] e(l) mod Jk+1)

(7.1.10) fk = fk(¯e(k)) =e ¯(k) ⊗ e¯(k) ⊗ 1 =e ˆ(k)

Thus, the q-Brauer algebra is cellularly stratiﬁed.

7.2 Consequences

In the following, we quote two important results of cellularly stratiﬁed algebra from [19] and apply them to the q-Brauer algebra. a) Cellularly stratiﬁed algebras and standard systems We recall the notion of a standardizable set by Dlab and Ringel ([9], Section 5) - which is called a standard system in [19]- of objects in an abelian category, given here for a module category:

Deﬁnition 7.2.1. Let A be any algebra, and suppose that we are given a ﬁnite set C of non-isomorphic modules C(k), indexed by k ∈ I, where I is endowed with a partial order ≤. Then the modules C(k) are said to form a standard system if the following three conditions hold:

(i) For all k ∈ I, EndA(C(k)) is a division ring. (ii) For all k, l ∈ I, If HomA(C(k),C(l)) 6= 0 then k ≥ l. 1 (iii) For all k, l ∈ I, If ExtA(C(k),C(l)) 6= 0 then k > l. Theorem 7.2.2. ( Hartmann, Henke, Koenig and Paget [19]). Let A be a cellularly stratiﬁed algebra with stratiﬁcation data (V1,B1,...,Vn,Bn).

1. Then the Specht modules of A form a standard system if and only if

for each k the Specht modules of Bk form a standard system.

2. Assume that for each k the Specht modules of Bk form a standard system. Then an A-module with a Specht filtration has well-defined filtration multiplicities.

The main results in this section are the following.

Theorem 7.2.3. Let Brn(N) be the q-Brauer algebra over a field R with [N] 6= 0. Then its Specht modules form a standard system if e(q) ≥ 4. In this case, modules with Specht filtrations have well-defined filtration multiplicities. 7.2. Consequences 85

Proof. Theorem 7.1.7 implies that the q-Brauer algebra Brn(N) is cellularly stratiﬁed with stratiﬁcation data

∗ ∗ (R,R,Hn(q),U1 ,U1,H3,n(q),...,U[n/2],U[n/2],H2[n/2]+1(q)).

Let k be an interger, 0 ≤ k ≤ [n/2]. For an index k if e(q) ≥ 4, then by Lemmas 1.3.17 and 1.3.18 the Specht modules Sλ of the Hecke algebra

H2k+1,n(q) of the symmetric group form a standard system with the usual dominance order on partitions λ of n − 2k. Now, applying Theorems 7.1.7 and 7.2.2 yield the precise result.

b) Young modules and Schur algebras for the q-Brauer algebras Hartmann, Henke, Koenig and Paget [19] showed that Young modules and Schur algebras can be defined for cellularly stratified algebras. Fur- ther, Schur - Weyl duality holds between a cellularly stratified algebra and its corresponding Schur algebra. For more detail we refer the reader to ([19], Sections 11-13). These results can be stated for the q-Brauer algebra

Brn(N) as follows.

For k integer, 0 ≤ k ≤ [n/2], and λ a partition of n − 2k, let Brn(N) be the q-Brauer algebra with the index set Λn endowed a partial order ¤ in Deﬁnition 6.1.7(2). The Specht modules of Brn(N) are denoted by C(k, λ). Denote Ypr(λ) the Young modules of the q-Brauer algebra Brn(N). Let aλ S(Brn(N)) = EndBrn(N)(⊕(Ypr(k, λ)) ) where the sum runs through all indices (k, λ) ∈ Λn and where aλ is chosen to equal the dimension of D(k, λ) if such a simple Brn(N)-module exists, or equals 1 otherwise. Then S(Brn(N)) is called the q-Schur algebra corresponding to Brn(N).

Theorem 7.2.4. Let Brn(N) (with [N] 6= 0) be the q-Brauer algebra with the index set Λn and assume that e(q) ≥ 4. Then:

1. The q-Brauer algebra has a standard system of Specht modules, and multiplicities in the Specht ﬁltration are well-deﬁned.

2. There exists a quasi-hereditary algebra S(Brn(N)) with the same partially ordered index set Λn such that the following statements hold true:

(a) The category F (C) of Brn(N)-modules with Specht ﬁltrations is equivalent, as an exact category, to the category of ∆-ﬁltered

S(Brn(N))-modules, where ∆ is a standard S(Brn(N))-module.

(b) The category of Brn(N)-modules with Specht ﬁltrations has relative projective covers, the Young modules. The algebra

S(Brn(N)) is the endomorphism algebra of a direct sum of a complete set of relative projective objects in F (C). 86 The cellularly stratiﬁed structure of the q-Brauer algebra

(c) Schur-Weyl duality holds between Brn(N) and S(Brn(N)). The faithfully balanced bimodule aﬀording the double centralizer prop-

erty between Brn(N) and S(Brn(N)) is the direct sum of the Young modules.

Proof. This theorem is a special case of Theorem 13.1 in [19]. Bibliography

[1] J. Birman, H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc., 313(1) (1989), 249-273.

[2] R. Brauer, On algebras which are connected with the semisimple con- tinuous groups, Ann. of Math., 63 (1937), 854-872.

[3] W.P. Brown, An algebra related to the orthogonal group, Michigan Math. J., 3 (1955), 1-22.

n [4] W.P. Brown, The semisimplicity of ωf , Ann. of Math., 63(2) (1956), 324 - 335.

[5] V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univer- sity Press, 1994.

[6] E. Cline, B. Parshall, L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math., 391 (1988), 85-99.

[7] R. Dipper, G.D. James, Representations of Hecke algebras and general linear groups, Proc.London Math. Soc., 52(3) (1986), 20-52.

[8] R. Dipper, G.D. James, Blocks and idempotents of Hecke algebras and general linear groups, Proc. London Math. Soc., 54(3) (1987), 57 - 82.

[9] V. Dlab and C.M. Ringel, The module theoretic approach to quasi-hereditary algebras. In ”Representations of Algebras and Related Topics” (Kyoto, 1990). London Math. Soc. Lecture Note Ser., 168, 200- 224. Cambridge University Press, 1992.

[10] W. Doran, D. Wales, P. Hanlon, On the semisimplicity of the Brauer centralizer algebras, J. Algebra, 211 (1999), 647 - 685.

[11] N.T. Dung, Cellular structure of q-Brauer algebras, arXiv:1208.6424, (2012).

[12] J. Enyang, Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras, J. Comb. Theory, Ser. A 26 (2007), 291-341. 88

[13] A.M. Gavrilik, A.U. Klimyk, q-Deformed orthogonal and pseudo- orthogonal algebras and their representations, Lett. Math. Phys. 21(3) (1991), 215-220.

[14] M. Geck, Hecke algebras of ﬁnite type are cellular, Invent. Math., 169 (3) (2007), 501-517.

[15] J.J. Graham, G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1) (1996), 1-34.

[16] P. Hanlon, D. Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra, 121(2) (1989), 409 - 445.

[17] P. Hanlon, D. Wales, Computing the discriminants of Brauer’s centralizer algebras, Math. Comp., 54 (1990), 771 - 796.

[18] R. Hartmann, R. Paget, Young modules and ﬁltration multiplicities for Brauer algebras, Math. Z., 254 (2006), 333-357.

[19] R. Hartmann, A. Henke, S. Koenig, R. Paget,Cohomological stratiﬁ- cation of diagrams algebras, Math. Ann., 347 (2010), 765-804.

[20] D. Hemmer, D. Nakano, Specht ﬁltration for Hecke algebras of type A, J. Lond. Math. Soc., 69(2) (2004), 623-638.

[21] G.D. James, The representation theory of the symmetric groups, SLN, 682, Springer-Verlag, New York, 1978.

[22] M. Jimbo, A q-analog of U(gl (n+1)), Hecke Algebra and the Yang- Baxter Equation, Lett. Math. Phys., 11 (1986), 247-252.

[23] S. Koenig, C. C. Xi, On the structure of cellular algebras. In: Algebras and modules II (Geiranger 1996), CMS Conf. Proc. 24, Amer. Math. Soc., (1998), 365-386.

[24] S. Koenig, C. C. Xi, Cellular algebras: Inﬂations and Morita equiva- lences, J. Lond. Math. Soc., 60(2) (1999), 700-722.

[25] S. Koenig, C. C. Xi,When is a cellular algebra quasi-heredity?, Mat. Ann., 315 (1999), 281-293.

[26] S. Koenig, C. C. Xi, A characteristic free approach to Brauer algebras, Trans. Amer. Math. Soc., 353(4) (2000), 1489-1505.

[27] R. Leduc, A. Ram, A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras, Adv. Math., 125(1) (1997), 1-94. 89

[28] A. Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series. American Mathemati- cal Society, Providence, R.I., vol. 15, 1999.

[29] A.I. Molev, A new quantum analog of the Brauer algebra, Czechoslovak J. Phys., 53 (2003), 1073-1078.

[30] A.I. Molev, private communication.

[31] E. Murphy, The representations of Hecke algebras of type An, J.Algebra, 173(1) (1995), 97-121.

[32] J. Murakami, The Kauﬀman polynomials of links and representation theory, Osaka J. Math., 24(4) (1987), 745-758.

[33] B. Parshall, L. Scott, Derived categories, quasi-hereditary algebras and algebraic groups, in “Proceeding of the Ottawa-Moosonee Workshop in Algebra 1987”, Math. Lecture Note series, Carleton University, 1988.

[34] H. Rui, M. Si, Gram determinants and semisimplicity criteria for Birman-Wenzl Algebras, J. Reine Angew. Math., 631 (2009), 153-179.

[35] H. Rui, A criterion on the semisimple Brauer algebras, J. Comb. The- ory, Ser. A, 111 (2005), 78-88.

[36] H. Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math., 128 (1988), 173-193.

[37] H. Wenzl, Quantum groups and subfactors of type B, C, and D, Comm. Math. Phys., 133 (1990), 383-432.

[38] H. Wenzl, A q-Brauer algebra, arXiv:1102.3892v1.

[39] H. Wenzl, A q-Brauer algebra, J. Algebra, 358 (2012), 102-127.

[40] H. Wenzl, Quotients of representation rings, Represent. Theory, 15 (2011), 385-406.

[41] H. Wenzl, Fusion symmetric spaces and subfactors, Paciﬁc J. Math., 259(2) (2012), 483-510.

[42] C. C. Xi, Partition algebras are cellular, Compositio Math., 119(1) (1999), 99-109.

[43] C. C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154 (2000), 280-298.