Action on flag varieties: 2-dimensional case
type: Item
instance of: scholarly article
title: Action on flag varieties: 2-dimensional case
zbMATH Open document ID: 0788.14041
DOI: 10.1007/BF01264520
author: Dae San Kim
author: Myung Hwan Kim
author: Jae Moon Kim
published in: Geometriae Dedicata
publication date: 1993-05-16T00:00:00Z
review text: Let \(A\) be a finite dimensional commutative semisimple algebra over a field \(k\), and let \(V\) be a finitely generated \(A\)-module. We examine the action of the general linear group \(GL_ A(V)\) on the set of flags of \(k\)-subspaces of \(V\). Also, let \((V,B)\) be a finitely generated `symplectic module' over \(A\). We also investigate the action of the symplectic group \(Sp_ A(V,B)\) on the set of flags of \(B'\)-isotropic \(k\)-subspaces of \(V\), where \(B'=\varphi \circ B\) is the \(k\)-symplectic form induced by a nonzero \(k\)-linear map \(\varphi:A\to k\). Studied earlier were the cases of \(GL_ A(V)\) acting on the set of \(k\)-subspaces of \(V\) and of \(Sp_ A(V,B)\) acting on the set of \(B'\)-isotropic \(k\)- subspaces of \(V\). So this paper is a natural extension of the previous works. In both cases, the orbits are completely classified in terms of certain integer invariants provided that \(\dim_ k A=2\), from which one can determine the precise structure of orbits, compute the exact number of orbits and give typical representatives for each orbit, etc.
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 166878
zbMATH Keywords: action of the general linear group on flags
zbMATH Keywords: action of the symplectic group
zbMATH Keywords: orbits
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 166878