Murphy operators and the centre of the Iwahori-Hecke algebras of type \(A\)
type: Item
instance of: scholarly article
title: Murphy operators and the centre of the Iwahori-Hecke algebras of type \(A\)
zbMATH Open document ID: 0947.20004
DOI: 10.1023/A:1018604404327
author: Andrew Mathas
published in: Journal of Algebraic Combinatorics
publication date: 2000-06-28T00:00:00Z
review text: This paper studies the question of whether the centre of the Iwahori-Hecke algebra \({\mathcal H}_{R,q}({\mathcal S}_n)\) of the symmetric group is the set of symmetric polynomials in the Murphy operators. Dipper and James have conjectured that this is indeed the case for an arbitrary base ring \(R\) containing an arbitrary invertible element \(q\), and they have proved the conjecture for \(\mathcal H\) semisimple although their proof contains a gap for the general case. The current paper reduces the problem to the combinatorial problem of checking an orthogonality property for a certain (previously unknown) set of polynomials. The author also considers a stronger conjecture due to James regarding the determinant of a certain matrix of inner products and offers a conjecture for the inverse of the matrix.
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1334707
zbMATH Keywords: centres
zbMATH Keywords: Iwahori-Hecke algebras
zbMATH Keywords: symmetric groups
zbMATH Keywords: symmetric polynomials
zbMATH Keywords: Murphy operators
zbMATH Keywords: orthogonality
reviewed by: R. M. Green
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Blocks and Idempotents of Hecke Algebras of General Linear Groups
cites work: A q -Analogue of the Jantzen-Schaper Theorem
cites work: The idempotents of the symmetric group and Nakayama's conjecture
full work available at URL: a:1018604404327
OpenAlex ID: W1595286908
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description: scientific article; zbMATH DE number 1334707