Irreducible Specht modules for Iwahori-Hecke algebras of type \(B\).
type: Item
instance of: scholarly article
zbMATH Open document ID: 1266.20004
DOI: 10.1090/S1088-4165-2012-00412-5
publication date: 2012-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.5660
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 6043199
zbMATH Keywords: Iwahori-Hecke algebras
zbMATH Keywords: irreducible Specht modules
zbMATH Keywords: cellular bases
zbMATH Keywords: tensor products of Hecke algebras
author: Matthew Fayers
MaRDI profile type: Publication
arXiv ID: 1005.5660
cites work: On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\)
cites work: Proof of the modular branching rule for cyclotomic Hecke algebras.
cites work: The number of simple modules of the Hecke algebras of type \(G(r,1,n)\)
cites work: The representation type of Hecke algebras of type \(B\).
cites work: On the tensor product of two basic representations of \(U_v(\hat s l_e)\)
cites work: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔤𝔩(𝔪|𝔫)
cites work: Representations of shifted Yangians and finite 𝑊-algebras
cites work: Graded decomposition numbers for cyclotomic Hecke algebras.
cites work: Highest weight categories arising from Khovanov's diagram algebra I: cellularity
cites work: Highest weight categories arising from Khovanov's diagram algebra. III: Category \(\mathcal O\)
cites work: Representations of Hecke algebras of type \(B_n\)
cites work: Hecke Algebras of Type Bn at Roots of Unity
cites work: Morita equivalences of Ariki-Koike algebras.
cites work: Reducible Specht modules.
cites work: Irreducible Specht modules for Hecke algebras of type \(A\).
cites work: Weights of multipartitions and representations of Ariki-Koike algebras.
cites work: Weights of multipartitions and representations of Ariki-Koike algebras. II: Canonical bases
cites work: On the irreducible Specht modules for Iwahori-Hecke algebras of type \(A\) with \(q=-1\).
cites work: Some reducible Specht modules for Iwahori-Hecke algebras of type \(A\) with \(q=-1\).
cites work: The reducible Specht modules for the Hecke algebra \(\mathcal H_{\mathbb C,-1}(\mathfrak S_n)\).
cites work: Cellular algebras
cites work: Strong multiplicity one theorems for affine Hecke algebras of type \(A\).
cites work: Rouquier blocks.
cites work: A q -Analogue of the Jantzen-Schaper Theorem
cites work: The Irreducible Specht Modules in Characteristic 2
cites work: The Jantzen sum formula for cyclotomic 𝑞–Schur algebras
cites work: Constructible characters and canonical bases
cites work: Q4380003
cites work: Some reducible Specht modules.
cites work: Some q-analogues of the Carter-Payne theorem
cites work: Blocks of cyclotomic Hecke algebras.
cites work: Q4378362
cites work: Q4699474
cites work: Tilting modules for cyclotomic Schur algebras
cites work: A Specht filtration of an induced Specht module.
Recommended article: Some reducible Specht modules for Iwahori-Hecke algebras of type \(A\) with \(q=-1\).
Recommended article: Cellular bases for the Brauer and Birman-Murakami-Wenzl algebras.
Recommended article: Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras.
Recommended article: Irreducible Specht modules for Hecke algebras of type \(A\).
Recommended article: Hecke Algebras with Unequal Parameters
Recommended article: On the irreducible Specht modules for Iwahori-Hecke algebras of type \(A\) with \(q=-1\).
Recommended article: Q4699474
Recommended article: Decomposable Specht modules for the Iwahori-Hecke algebra \(\mathcal H_{\mathbb F,-1}(\mathfrak S_n)\).
Recommended article: Crystal of affine \(\widehat{\mathfrak{sl}}_\ell\) and modular branching rules for Hecke algebras of type \(D_n\)
Recommended article: OCNEANU'S TRACE AND STARKEY'S RULE
title: Irreducible Specht modules for Iwahori-Hecke algebras of type \(B\).
published in: Representation Theory
review text: Let \(\mathcal B_n\) denote the Iwahori-Hecke algebra of type \(B_n\) with parameters \(Q\) and \(q\) defined over a field \(\mathbb F\). This algebra has a cellular basis in the sense of \textit{J. J. Graham} and \textit{G. I. Lehrer} [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] with Specht modules \(S^{(\lambda,\mu)}\) indexed by the bipartitions of \(n\). This paper studies when these Specht modules are irreducible.NEWLINENEWLINE In the case when \(-Q\) is not a power of \(q\) in \(\mathbb F\), the author uses a result of \textit{R. Dipper} and \textit{G. James} [J. Algebra 146, No. 2, 454-481 (1992; Zbl 0808.20016), Theorem 4.17] that the algebra \(\mathcal B_n\) is Morita equivalent to a sum of tensor products of Hecke algebras of type \(A\) under a map sending the Specht module \(S^{(\lambda,\mu)}\) for \(\mathcal B_n\) to the tensor product \(S^\lambda\otimes S^\mu\) of Specht modules for Hecke algebras of type \(A\). This reduces to the type \(A\) case, where a complete classification of reducible Specht modules is available in all cases except \(q=-1\).NEWLINENEWLINE Section 3 of the paper gives a complete classification in the case where \(Q=-q^r\) where \(r\in\mathbb N_0\) and \(q\) is not a root of unity. Section 4 gives some partial results on the case \(q=-1\) and \(Q=\pm 1\) including a sufficient condition for \(S^{(\lambda,\mu)}\) to be reducible, stated in terms of the number of addable and removable \(1\) and \(-1\) nodes in the partitions \(\lambda\) and \(\mu\). Moreover, a reduction is given to the analogous problem for type \(A\) Hecke algebras when \(q=-1\).
reviewed by: Mark Wildon
description: scientific article; zbMATH DE number 6043199