Hilbert series of group representations and Gröbner bases for generic modules
type: Item
instance of: scholarly article
title: Hilbert series of group representations and Gröbner bases for generic modules
zbMATH Open document ID: 0821.13011
DOI: 10.1023/A:1022445607540
author: Shmuel Onn
published in: Journal of Algebraic Combinatorics
publication date: 1995-09-25T00:00:00Z
review text: Let \(G\) be a finite group, \(V\) an \(n\)-dimensional vector space over an algebraically closed field \(K\) whose characteristic does not divide the order of \(G\), \(A\) the \(K\)-algebra of polynomial functions from \(V\) to \(K\), and let \(\pi : G \to GL_ K (V)\) be a representation of \(G\). The set of polynomial maps from \(V\) to \(V\) is a graded \(n\)-dimensional free module over \(A\) and contains \(\pi (G)\). In this paper \(M (\pi)\), the \(A\)- submodule generated by \(\pi (G)\), is studied. First it is shown that the decomposition of \(\pi\) into irreducible representations induces a decomposition of \(M (\pi)\) into generic modules. This implies a formula for the rank and the first Betti number of \(M (\pi)\). Then, applying Gröbner basis methods to generic modules, the author computes the second Betti number and the Hilbert series for \(M (\pi)\).
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 20C40
zbMATH DE Number: 558524
zbMATH Keywords: linear representations of finite groups
zbMATH Keywords: set of polynomial maps
zbMATH Keywords: Gröbner basis
zbMATH Keywords: second Betti number
zbMATH Keywords: Hilbert series
reviewed by: F. Pauer
MaRDI profile type: Publication
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full work available at URL: a:1022445607540
OpenAlex ID: W1598550888
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description: scientific article; zbMATH DE number 558524