Multiplicities of Schur functions in invariants of two \(3\times 3\) matrices
type: Item
instance of: scholarly article
title: Multiplicities of Schur functions in invariants of two \(3\times 3\) matrices
zbMATH Open document ID: 1027.16013
DOI: 10.1016/S0021-8693(03)00070-X
author: Georgi K. Genov
author: Vesselin Drensky
published in: Journal of Algebra
publication date: 2003-07-30T00:00:00Z
review text: With a symmetric power series \(f(x,y)\) the authors associate its multiplicity series\break \(\sum m(\lambda_1,\lambda_2)t^{\lambda_1}u^{\lambda_2}\), where \(f(x,y)=\sum m(\lambda_1,\lambda_2)S_{(\lambda_1,\lambda_2)}(x,y)\) expanded as a series of Schur functions. When \(f(x,y)\) is the Hilbert (or Poincaré) series of a polynomial module over \(\text{GL}_2\), the complex general linear group, the number \(m(\lambda_1,\lambda_2)\) is the multiplicity as a direct summand of the irreducible \(\text{GL}_2\)-module indexed by the partition \((\lambda_1,\lambda_2)\). The Hilbert series of the algebra of simultaneous conjugation invariants of pairs of \(3\times 3\) matrices is known from the work of \textit{Y. Teranishi} [Nagoya Math. J. 104, 149-161 (1986; Zbl 0615.16013)]. The main result of the present paper is an explicit rational expression for the multiplicity series of this Hilbert series. As a consequence, the asymptotics of the multiplicities in the trace cocharacter sequence of pairs of \(3\times 3\) matrices is determined. Along the way the authors express as a rational function the multiplicity series of the symmetric tensor algebra of the third symmetric tensor power of the vector representation of \(\text{GL}_2\). This is in the spirit of a classical result of \textit{R. M. Thrall} [Am. J. Math. 64, 371-388 (1942; Zbl 0061.04201)] on certain plethysms.
reviewed by: Matyas Domokos
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1956747
zbMATH Keywords: matrix invariants
zbMATH Keywords: trace rings
zbMATH Keywords: symmetric functions
zbMATH Keywords: Schur functions
zbMATH Keywords: Hilbert series
zbMATH Keywords: irreducible modules
zbMATH Keywords: conjugation invariants
zbMATH Keywords: cocharacter sequences
MaRDI profile type: Publication
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full work available at URL: s0021-8693(03)00070-x
OpenAlex ID: W1989341273
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description: scientific article; zbMATH DE number 1956747