A generalization of Weyl's denominator formulas for the classical groups.
type: Item
instance of: scholarly article
title: A generalization of Weyl's denominator formulas for the classical groups.
zbMATH Open document ID: 1108.20045
author: Masahiko Ito
author: Kazuhiko Koike
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Let \(1<k<n\). We consider an \(n\choose k\) by \(n\choose k\) matrix \(M\) whose rows are indexed by the Young diagrams that fit inside the diagram \((n-k)^k\) and whose columns are indexed by \(k\)-element subsets \(\{i_1,\dots,i_k\}\) of \(\{1,\dots,n\}\). The entry at the intersection of the column whose index is \(\lambda\) with the row of index \(\{i_1,\dots,i_k\}\) is defined to be \(s_\lambda(x_{i_1},\dots,x_{i_k})\), where \(s_\lambda\) is a Schur function and \(x_1,\dots,x_n\) are variables. One also orders the rows and columns appropriately. The first theorem is then that the determinant of \(M\) is a power of the Weyl denominator for \(\text{GL}_n\). For instance, if \(k=1\), we have \(s_\lambda(x_i)=x_i^{|\lambda|}\) and one gets the usual expression of the Weyl denominator as a Vandermonde determinant. The Schur functions are characters of general linear groups and thus have analogues for symplectic groups, orthogonal groups, and spin groups. The authors prove in all these cases that the determinant is a power of the Weyl denominator.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 5078149
zbMATH Keywords: Vandermonde determinants
zbMATH Keywords: Weyl denominators
zbMATH Keywords: Young diagrams
zbMATH Keywords: Schur functions
zbMATH Keywords: characters of general linear groups
zbMATH Keywords: symplectic groups
zbMATH Keywords: orthogonal groups
zbMATH Keywords: spin groups
Wikidata QID: Q112881494
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.026
OpenAlex ID: W2012041484
cites work: BCn-type Jackson integral generalized from Gustafson'sCn-type sum
cites work: A determinant formula for a holonomic \(q\)-difference system associated with Jackson integrals of type \(BC_n\)
cites work: Q4002278
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Representations of spinor groups and the difference characters of \(SO(2n)\)
cites work: Q3947818
cites work: Q4435456
DOI: 10.1016/J.JALGEBRA.2006.05.026
description: scientific article; zbMATH DE number 5078149