Linear preserver problems and algebraic groups
type: Item
instance of: scholarly article
title: Linear preserver problems and algebraic groups
zbMATH Open document ID: 0836.20065
DOI: 10.1007/BF01460985
published in: Mathematische Annalen
publication date: 1996-02-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165363
review text: Let \(M_n=M_n(K)\) be the space of \(n\) by \(n\) matrices over an algebraically closed field \(K\) of characteristic 0. We have a direct decomposition \(M_n=M^0_n\oplus K\cdot 1\), where 1 denotes the identity matrix, and \(M^0_n\) the subspace defined by \(\text{tr}(x)=0\). We denote by \(\text{GL}(n^2)\) the group of vector space automorphisms of \(M_n\) and by \(\text{GL}(n^2-1)\) its subgroup which fixes 1 and preserves \(M^0_n\). By \(G\) we denote the subgroup consisting of all similarity transformations \(x\mapsto axa^{-1}\) where \(a\in\text{GL}(n)\). Note that \(G\cong\text{PSL}(n)\). We define \(\tau\in\text{GL}(n^2-1)\) to be the transposition map and we write \(x'\) for \(\tau(x)\). We classify all algebraic subgroups \(\Gamma\) of \(\text{GL}(n^2)\) containing \(G\). Let \(O(n^2)\) be the subgroup of \(\text{GL}(n^2)\) which preserves the non-degenerate symmetric bilinear form \(\text{tr}(xy) \), \(x, y\in M_n\), \(O(n^2-1)\) its subgroup consisting of operators which fix the identity matrix 1. Let \(P\) (resp. \(Q\)) be the subgroup of \(\text{GL}(n^2)\) which acts trivially on \(M^0_n\) and \(M_n/M^0_n\) (resp. \(K\cdot 1\) and \(M_n/K\cdot 1\)). By \(T\) we denote the 2-dimensional subtorus of \(\text{GL}(n^2)\) which acts by scalar transformations on \(M^0_n\) and \(K \cdot 1\), and we set \(T_1=T\cap\text{SL}(n^2)\). If \(n=4\), there is an overgroup \(\Lambda\cong\text{SL}(6)/\langle -1\rangle\) of \(G\) in \(\text{SL}(15)\). Theorem A. Let \(\Gamma\) be a proper connected algebraic subgroup of \(\text{SL}(n^2)\) containing \(G=\text{PSL}(n)\). Then (i) if reducible, \(\Gamma\) is one of the groups \(H\), \(HP\), \(HQ\), \(HT_1\), \(HPT_1\), \(HQT_1\), where \(H=G\), \(\text{SO}(n^2-1)\), \(\text{SL}(n^2-1)\), or \(\Lambda\) if \(n=4\); (ii) if irreducible, \(\Gamma\) is one of the groups \(t\text{SO}(n^2)t^{-1}\), \(t(\text{SL}(n)\otimes\text{SL}(n)^{\text{opp}})t^{-1}\), \(t\in T_1\). For applications to linear preserver problems it is necessar…
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 840180
zbMATH Keywords: space of \(n\) by \(n\) matrices
zbMATH Keywords: group of vector space automorphisms
zbMATH Keywords: similarity transformations
zbMATH Keywords: algebraic subgroups
zbMATH Keywords: symmetric bilinear forms
zbMATH Keywords: linear preserver problems
zbMATH Keywords: normalizers
author: V. P. Platonov
author: Dragomir Ž. Đoković
MaRDI profile type: Publication
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OpenAlex ID: W2084906853
description: scientific article; zbMATH DE number 840180