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The Q-Brauer Algebras The q-Brauer algebras Von der Fakult¨atMathematik und Physik der Universit¨atStuttgart zur Erlangung der W¨urdeeines 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 ¨uber einem kommutativen Ring mit Einselement oder einem K¨orper der Charakteristik p ≥ 0. Wir konstruieren zun¨achst eine Zell-asis f¨urdie q-Brauer Algebra ¨uber einem kommutativen Ring mit Einselement und zeigen dann, dass die q-Brauer Algebra zellul¨ar ist im Sinne von Graham und Lehrer. Anschließend klassifizieren wir alle einfachen Moduln bis auf Isomorphie f¨urden Fall der q-Brauer Algebra ¨uber einem K¨orper 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 k¨onnen. Im Falle der q-Brauer Algebra ¨uber einem K¨orper beliebiger Charakteristik bestimmen wir weiterhin die Werte des Parameters, f¨ur die die q-Brauer Algebra quasi-erblich ist. Außer- dem zeigen wir, dass die q-Brauer Algebra zellul¨ar stratifiziert ist durch Angabe einer entsprechenden iterierten Aufblasungsstruktur. Dies wiederum impliziert Ergebnisse ¨uber die Eindeutigkeit von Specht- Filtrierungsvielfachheiten der q - Brauer Algebra, sowie die Existenz von Young Moduln und Schur Algebren f¨ur 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 dia- grams of the classical Brauer algebra, and is helpful for producing cellular structure of the q-Brauer algebra.
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