Hilbert series of fixed free algebras and noncommutative classical invariant theory
type: Item
instance of: scholarly article
title: Hilbert series of fixed free algebras and noncommutative classical invariant theory
zbMATH Open document ID: 0628.16004
DOI: 10.1016/0021-8693(85)90183-8
author: Gert Almkvist
author: Warren Dicks
author: Edward Formanek
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G be a group acting linearly on a finite-dimensional vector space V over a field of characteristic 0. The authors study the invariants \(T(V)^ G\) of G on the tensor algebra T(V). They prove a Molien-Weyl- type formula for the Poincaré series \(H(T(V)^ G)\) for a compact G (generalizing the earlier result of the last two authors for finite groups [see Linear Multilinear Algebra 12, 21-30 (1982; Zbl 0493.15020)]. They describe \(H(T(V)^ G)\) for a polynomial representation of \(G=SL_ r\) on V restricted to the group of unipotent \(r\times r\) upper triangular matrices. They show that \(H(T(V))^{SL_ 2}\), for a polynomial representation of \(SL_ 2\) on V, is algebraic over Z[t]. Finally, they develop a theory of invariants of a binary form with noncommuting coefficients analogous to the classical invariant theory of Sylvester and Franklin, and compute an asymptotic formula for the number of invariants of a given degree.
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 4021372
zbMATH Keywords: invariants
zbMATH Keywords: tensor algebra
zbMATH Keywords: Molien-Weyl-type formula
zbMATH Keywords: Poincaré series
zbMATH Keywords: polynomial representation
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90183-8
OpenAlex ID: W2032739988
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description: scientific article; zbMATH DE number 4021372