A conjecture of Procesi and a new basis for the decomposition of the graded left regular representation of \(S_ n\)
type: Item
instance of: scholarly article
title: A conjecture of Procesi and a new basis for the decomposition of the graded left regular representation of \(S_ n\)
zbMATH Open document ID: 0795.20006
author: Edward E. Allen
published in: Advances in Mathematics
publication date: 1994-09-18T00:00:00Z
review text: Let \(\mathbb{R}^*\) denote the quotient of the ring of polynomials in \(n\) variables over the ideal of symmetric polynomials in these variables. The graded character \(\text{char}_ q \mathbb{R}^*\) decomposes into irreducible constituents by \[ \text{char}_ q \mathbb{R}^*= \sum_ \lambda \chi^ \lambda \sum_ S q^{| C(S)|}, \] where \(\chi^ \lambda\) denotes the irreducible character indexed by the partition \(\lambda\), \(S\) ranges over all standard tableaux of shape \(\lambda\), and \(| C(S)|\) is the sum of the entries in the cocharge tableau obtained from \(S\). The author finds a basis for \(\mathbb{R}^*\) that has natural triangularity and explicitly exhibits this decomposition. Furthermore, when this basis is restricted to \(\mathbb{R}_ \mu\), one of a family of \(S_ n\)-modules recently studied by Garsia and Procesi, it yields a basis for this module. This then suggests a new approach to the charge interpretation for the \(q\)-Kostka-Foulkes polynomials \(K_{\lambda\mu} (q)\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 490809
zbMATH Keywords: Kostka-Foulkes polynomials
zbMATH Keywords: symmetric polynomials
zbMATH Keywords: graded character
zbMATH Keywords: irreducible constituents
zbMATH Keywords: irreducible character
zbMATH Keywords: partition
zbMATH Keywords: standard tableaux
zbMATH Keywords: cocharge tableau
zbMATH Keywords: \(S_ n\)-modules
zbMATH Keywords: charge interpretation
reviewed by: David M. Bressoud
MaRDI profile type: Publication
full work available at URL: aima.1993.1035
OpenAlex ID: W2002574151
Wikidata QID: Q123143275
DOI: 10.1006/AIMA.1993.1035
description: scientific article; zbMATH DE number 490809