Cauchy decompositions and invariants
type: Item
instance of: scholarly article
title: Cauchy decompositions and invariants
zbMATH Open document ID: 0985.20030
author: Mihalis Maliakas
published in: Mathematische Zeitschrift
publication date: 2002-02-17T00:00:00Z
review text: Let \(k\) be an algebraically closed field. Given a pair of classical algebraic groups \((G_1,G_2)\) contained in \(\text{GL}_n\) and \(\text{GL}_m\) over \(k\) respectively, let \(X(G_1,G_2)\) be the affine variety of \(n\times m\)-matrices \(M\) such that \(M^tJ_{G_1}M=0\) and \(MJ_{G_2}M^t=0\), where \(J_{G_1}\) (resp. \(J_{G_2}\)) is the defining matrix for \(G_1\) (resp. \(G_2\)) in \(\text{GL}_n\) (resp. \(\text{GL}_m\)). \(X(G_1,G_2)\) is naturally a \(G_1\times G_2\) variety and its coordinate algebra \(A(G_1,G_2)\) is a graded algebra on which \(G_1\times G_2\) acts homogeneously. The question is whether \(A(G_1,G_2)\) has a good filtration as a \(G_1\times G_2\)-module (i.e., each homogeneous component has a good filtration), what are the factors. This is a generalization of Cauchy decomposition and the standard basis theorem for general linear groups to other classical groups. One of the main results is that, under certain restrictions on the appearance of a group of type \(D_n\) and the characteristic, \(A(G_1,G_2)\) has a good filtration with subquotients isomorphic to \(\nabla_{G_1}(\lambda^*)\otimes\nabla_{G_2}(\lambda)\), where the weights \(\lambda\) are described as partitions with number of parts at most \(\min(\text{rk}(G_1),\text{rk}(G_2))\) and \(\lambda^*=-w_0(\lambda)\) for the longest element \(w_0\) of the Weyl group element for \(G_1\). The second main result is an application of the first main result to the case \(G_2=\text{GL}_n\) and \(G_1\) arbitrary, to give a set of generators of the algebra \(A(G_1,G_2)^H\) of \(H\)-invariants with \(H\) being a subgroup of \(\text{GL}_n\) of the forms \(\text{SL}_n\), \(\text{Sp}_n\) (for \(n\) even), and \(\text{SO}_n\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 1559187
zbMATH Keywords: algebraic groups
zbMATH Keywords: affine varieties
zbMATH Keywords: good filtrations
zbMATH Keywords: Cauchy decompositions
zbMATH Keywords: standard bases
zbMATH Keywords: weights
zbMATH Keywords: generators
zbMATH Keywords: algebras of invariants
reviewed by: Zongzhu Lin
MaRDI profile type: Publication
full work available at URL: s002090000146
OpenAlex ID: W1998618962
DOI: 10.1007/S002090000146
description: scientific article; zbMATH DE number 1559187