The functional equation for Poincaré series of trace rings of generic 2\(\times 2\) matrices
type: Item
instance of: scholarly article
title: The functional equation for Poincaré series of trace rings of generic 2\(\times 2\) matrices
zbMATH Open document ID: 0587.16014
DOI: 10.1007/BF02774086
author: Lieven Le Bruyn
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(G_{m,n}\) be the algebra generated by m generic \(n\times n\) matrices. By adjoining to \(G_{m,n}\) the traces of all its elements one obtains the trace ring \(\Pi_{m,n}\); \(G_{m,n}\) and \(\Pi_{m,n}\) play important roles in the theory of the algebras with polynomial identities. In the paper under review the trace ring of \(2\times 2\) matrices is studied. Using some 60-year-old results due to I. Schur and H. Weyl the author gives a rational expression of the Poincaré (or Hilbert) series \({\mathcal P}(\Pi_{m,2},t_ 1,...,t_ m)\) of the graded vector space \(\Pi_{m,2}\). Up till now the explicit form for these series has been known for \(m\leq 4\) only [\textit{E. Formanek}, J. Algebra 89, 178-223 (1984; Zbl 0549.16008)]. As a consequence the author establishes that the Poincaré series of \(\Pi_{m,2}\) satisfies the functional equation \[ {\mathcal P}(\Pi_{m,2},1/t)=-t^{4m}{\mathcal P}(\Pi_{m,2},t),\quad m>2. \] Recently \textit{E. Formanek} [Trans. Am. Math. Soc. 294, 647-663 (1986)] and \textit{Y. Teranishi} [Nagoya Math. J. 104 (1986) (to appear)] independently have generalized this result. They have obtained that the trace ring of \(n\times n\) generic matrices satisfies a similar functional equation. A complete exposition on various properties of \(\Pi_{m,2}\) can be found in [\textit{L. Le Bruyn}, Trace rings of generic 2 by 2 matrices, Mem. Am. Math. Soc. (to appear)].
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 3941736
zbMATH Keywords: trace ring
zbMATH Keywords: algebras with polynomial identities
zbMATH Keywords: graded vector space
zbMATH Keywords: Poincaré series
zbMATH Keywords: functional equation
zbMATH Keywords: generic matrices
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02774086
OpenAlex ID: W2026297380
description: scientific article; zbMATH DE number 3941736