Hilbert series of PI relatively free \(G\)-graded algebras are rational functions.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1259.16023
author: Eli Aljadeff
author: A. Ya Belov
published in: Bulletin of the London Mathematical Society
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.3253
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 6048037
zbMATH Keywords: Hilbert series
zbMATH Keywords: rationality
zbMATH Keywords: group graded algebras
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: graded T-ideals
zbMATH Keywords: PI algebras
MaRDI profile type: Publication
OpenAlex ID: W3105536892
arXiv ID: 1011.3253
DOI: 10.1112/BLMS/BDR116
title: Hilbert series of PI relatively free \(G\)-graded algebras are rational functions.
review text: Let \(F\) be a field of characteristic 0 and let \(G\) be a finite group. Consider the free associative \(G\)-graded algebra \(F(X;G)\) on \(r\) homogeneous (in the \(G\)-grading) free generators. Take an ideal \(I\) in \(F(X;G)\) that is invariant under all endomorphisms that respect the \(G\)-grading. Such ideals are called graded T-ideals; these are the ideals of \(G\)-graded identities of a \(G\)-graded algebra. Suppose further that the relatively free graded algebra \(F(X;G)/I\) satisfies an ordinary polynomial identity.NEWLINENEWLINE The main theorem of the paper under review proves that the Hilbert series (in one variable) of \(F(X;G)/I\) is a rational function. Moreover the authors prove that the same holds if one considers the multivariate Hilbert series of \(F(X;G)/I\). Another result worth mentioning is that the Hilbert series of each \(G\)-homogeneous component of \(F(X;G)/I\) is also a rational function.NEWLINENEWLINE The proofs of these results are quite involved and will be of significant interest to the specialists in PI algebras.
reviewed by: Plamen Koshlukov
description: scientific article; zbMATH DE number 6048037