Representations of G-Brauer Algebras

Parvathi, M.; Savithri, D.

doi:10.1007/s10012-002-0453-6

Cited by 7 publications

(15 citation statements)

References 4 publications

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“…Finally, we remark that other variants of signed and cyclotomic Brauer algebras, G-Brauer algebras and cyclotomic BMW have been studied previously in the papers [CGW05], [GH], [HO01], [PK98], [PK02], [PS02], [RY04]. In [Naz96], Nazarov introduced an affine analogue of the Brauer algebra which he called the (degenerate) affine Wenzl algebra.…”

Section: §1 Introductionmentioning

confidence: 98%

Cyclotomic Nazarov-Wenzl Algebras

Ariki¹,

Mathas²,

Rui³

2006

Nagoya Mathematical Journal

256

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Abstract. Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain "cyclotomic quotients" of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r n (2n − 1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

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Section: §1 Introductionmentioning

confidence: 98%

Cyclotomic Nazarov-Wenzl Algebras

Ariki¹,

Mathas²,

Rui³

2006

Nagoya Mathematical Journal

256

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“…Thus the A-Brauer algebra D n (A) is free as an R-module with basis the set of unoriented Glabeled n-strand Brauer diagrams. These G-Brauer algebras are the same as those defined by Parvathi and Savithri [23].…”

Section: 5mentioning

confidence: 83%

Cellularity of wreath product algebras and A-Brauer algebras

Geetha

Goodman

2013

Journal of Algebra

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Abstract. A cellular algebra is called cyclic cellular if all cell modules are cyclic. Most important examples of cellular algebras appearing in representation theory are in fact cyclic cellular. We prove that if A is a cyclic cellular algebra, then the wreath product algebras A ≀ Sn are also cyclic cellular. We also introduce A-Brauer algebras, for algebras A with an involution and trace. This class of algebras includes, in particular, G-Brauer algebras for non-abelian groups G. We prove that if A is cyclic cellular then the A-Brauer algebras Dn(A) are also cyclic cellular. IntroductionThe concept of cellularity of algebras was introduced by Graham and Lehrer [13]. Cellularity is useful for analyzing the representation theory of important classes of algebras such as Hecke algebras, q-Schur algebras, Brauer and BMW algebras, etc.In this paper, we introduce a variant of the notion of cellularity: A cellular algebra A is called cyclic cellular if all of its cell modules are cyclic A-modules. Although cyclic cellularity is nominally stronger than cellularity, most important classes of cellular algebras appearing in representation theory are in fact cyclic cellular. For example, the algebras mentioned above -Hecke algebras of type A, q-Schur algebras, Brauer algebras and BMW algebrasare cyclic cellular.It appears that the requirement of cyclic cell modules eliminates some potential pathology allowed by the general notion of cellularity; for example, an abelian cellular algebra is cyclic cellular if and only if all the cell modules have rank 1.Our main theorem regarding cyclic cellular algebras is the following: if A is a cyclic cellular algebra, then the wreath product of A with the symmetric group S n is also cyclic cellular.In the second part of the paper, we introduce a new class of algebras, the A-Brauer algebras, which are a sort of wreath product of an algebra A with the Brauer algebra. The definition of the A-Brauer algebras requires that A be an algebra with involution * and 2010 Mathematics Subject Classification. 16G30. Key words and phrases. Cellular algebras, wreath products, Brauer algebras. We thank Andrew Mathas for suggesting that the method of [7] could be adapted to prove the cellularity of wreath products. T. Geetha was supported by a postdoctoral fellowship at

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“…The Proof is similar as in [9] and [15] Let J i = W r,s e r,s,i W r,s then we get filtration of ideals…”

Section: Walled Cyclic G-brauer Algebramentioning

confidence: 94%

“…In this paper, we generalize the algebra to walled cyclic G-Brauer algebra and study its generators, relations, cellularity and the necessary and sufficient condition for cyclic G-Brauer algebra to be quasi heriditary. Walled cyclic G-Brauer algebra is an algebra consisting of walled cyclic G-Brauer diagrams which are subalgebra of cyclic G-Brauer algebra introduced by [15].…”

Section: Introductionmentioning

confidence: 99%