On automorphisms of matrix invariants induced from the trace ring
type: Item
instance of: scholarly article
title: On automorphisms of matrix invariants induced from the trace ring
zbMATH Open document ID: 0802.16017
DOI: 10.1016/0024-3795(93)90271-O
author: Zinovy Reichstein
published in: Linear Algebra and its Applications
publication date: 1994-04-27T00:00:00Z
review text: Assume \(M^ m_ n\) is the affine space of all \(m\)-tuples of \(n\) by \(n\) matrices over a fixed algebraically closed field of characteristic zero and let the group \(PGL_ n\) act on this space by (simultaneous) conjugations. Denote the corresponding quotient space by \(Q_{m,n}\), and \(A = \text{Aut}(Q_{m,n})\) be the group of automorphisms of \(Q_{m,n}\) that preserve the representation type of \(Q_{m,n}\). The main results proved in the paper under review are the following. (i) If \(m \geq n + 1\) then for any positive integer \(s\) the group \(A\) acts \(s\)-transitively on \(Q_{m,n}\) (considered as a stratified affine variety). (ii) Let \(T_{m,n}\) be the trace ring of the ring of \(m\) generic \(n\) by \(n\) matrices. Then there exist elements in \(A\) (for \(n \geq 3\) and \(m \geq 2\)) that cannot be induced by any automorphism of \(T_{m,n}\).
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 539965
zbMATH Keywords: group of automorphisms
zbMATH Keywords: representation type
zbMATH Keywords: stratified affine variety
zbMATH Keywords: trace ring
zbMATH Keywords: generic \(n\) by \(n\) matrices
Wikidata QID: Q117378936
MaRDI profile type: Publication
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full work available at URL: 0024-3795(93)90271-o
OpenAlex ID: W2043709146
description: scientific article; zbMATH DE number 539965