Hecke algebras and immanants
type: Item
instance of: scholarly article
title: Hecke algebras and immanants
zbMATH Open document ID: 0831.20008
DOI: 10.1016/0024-3795(94)90508-8
published in: Linear Algebra and its Applications
publication date: 1996-02-06T00:00:00Z
review text: Let \(R\) be an algebra and let \(e\in R\) be an idempotent. The algebra \(eRe\) is called the Hecke algebra associated with \(e\). The author discusses the case when \(R=\mathbb{C} G\); the group algebra of a finite group \(G\) over \(\mathbb{C}\) and \(e={\psi(1)\over|H|}\sum_{h\in H}\psi(h^{-1})h\) where \(H\leq G\) and \(\psi\) is a character of \(H\). Every irreducible character \(\chi\) of \(\mathbb{C} G\) can be restricted to \(H=e\mathbb{C} Ge\) to give (if non-zero) an irreducible character of \(H\). Conversely, every irreducible character of \(H\) is the restriction of some irreducible character of \(\mathbb{C} G\). The author describes how character values of \(H=e\mathbb{C} Ge\) can be used to give linear combinations of immanants which are positive on all Hermitian positive semidefinite matrices \(A\). As an application, a short proof is given of a theorem of \textit{G. D. James} and \textit{M. W. Liebeck} [Proc. Lond. Math. Soc., III. Ser. 55, 243-265 (1987; Zbl 0657.15006)] that the permanent of \(A\) is greater than the normalized immanants which correspond to two-part partitions.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 553980
zbMATH Keywords: idempotents
zbMATH Keywords: Hecke algebras
zbMATH Keywords: group algebras
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: immanants
zbMATH Keywords: Hermitian positive semidefinite matrices
zbMATH Keywords: permanents
author: Gordon James
reviewed by: Ahmed A. Khammash
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90508-8
OpenAlex ID: W2090774362
cites work: On the Degrees and Rationality of Certain Characters of Finite Chevalley Groups
cites work: Q3922829
cites work: Immant dominance orderings for hook partitions
cites work: Q4103728
cites work: The representation theory of the symmetric groups
cites work: Permanents and Immanants of Hermitian Matrices
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description: scientific article; zbMATH DE number 553980