An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants
type: Item
instance of: scholarly article
title: An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants
zbMATH Open document ID: 1167.13004
author: Takuya Ohta
published in: Hokkaido Mathematical Journal
publication date: 2008-10-20T00:00:00Z
review text: Let \(V\) be a finite-dimensional complex vector space and \(\sigma:\text{End}(V)\to \text{End}(V)\) be a \({\mathbb C}\)-linear anti-automorphism. Let \(\tilde G\) be a reductive subgroup of \(GL(V)\) acting on a subvector space \(\tilde L\) of \(\text{End}(V)\) by the adjoint action. Suppose that \(\tilde G\) is invariant under \(\sigma\) , and that the restriction of \(\sigma\) to \(\tilde G\) is an involution. Consider a closed subgroup \(G\) of \(\tilde G\) acting on a subspace \(L\) of \(\tilde L\). In this article, a sufficient condition on the inclusion \((G,L)\hookrightarrow (\tilde G, \tilde L)\) is given for which the orbit correspondence \(L/G\mapsto \tilde L/\tilde G\) (\({\mathcal O}\mapsto Ad(\tilde G)\cdot{\mathcal O}\)) is injective. Moreover, under these conditions, the ring \({\mathbb C}[L]^G\) of \(G\)-invariants on \(L\) is the integral closure of \({\mathbb C}[\tilde L]^{\tilde G}|_L\) in its quotient field. In particular, if \({\mathbb C}[\tilde L]^{\tilde G}|_L\) is normal, then the restriction map on the invariants is surjective. Examples are given for which the conditions are satisfied. It is shown, for example, that one can deduce the First Fundamental Theorem of Invariant Theory for the orthogonal and symplectic groups from that of the general linear group.
reviewed by: Lucy Moser-Jauslin
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5354266
zbMATH Keywords: invariant theory
zbMATH Keywords: first fundamental theorem of invariant theory
zbMATH Keywords: inclusion theorems between sets of orbits
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1253539529
OpenAlex ID: W1970063853
DOI: 10.14492/HOKMJ/1253539529
description: scientific article; zbMATH DE number 5354266