The four subspace problem: an elementary solution.
type: Item
instance of: scholarly article
title: The four subspace problem: an elementary solution.
zbMATH Open document ID: 1079.16007
author: Gonzalo Medina
author: Alexander Zavadskij
published in: Linear Algebra and its Applications
publication date: 2004-11-26T00:00:00Z
review text: Let \(K\) be an arbitrary field. We recall that a quadruple of \(K\)-vector spaces is a system \(U=(U_0,U_1,U_2,U_3,U_4)\), where \(U_0\) is a finite dimensional \(K\)-vector space and \(U_1,U_2,U_3,U_4\) are subspaces of \(U_0\). A morphism \(f\colon U\to U'\) between two quadruples \(U\) and \(U'\) is a linear map \(f\colon U_0\to U'_0\) such that \(f(U_j)\subseteq U'_j\), for \(j=1,2,3,4\). The morphism \(f\colon U\to U'\) is an isomorphism if the linear map \(f\colon U_0\to U'_0\) is bijective. We set \(U\oplus U'=(U_0\oplus U'_0,U_1\oplus U'_1,U_2\oplus U'_2,U_3\oplus U'_3,U_4\oplus U'_4)\), and we call \(V\) indecomposable if \(V\) is not isomorphic to a direct sum \(U\oplus U'\) of two non-zero quadruples. The Four Subspace Problem is the problem of classifying all indecomposable quadruples, up to isomorphism. It was solved over 30 years ago by Gelfand and Ponomarev (over an algebraically closed field), and independently by Nazarova (over an arbitrary field). The paper contains a new elementary and short proof of the problem. The endomorphism algebra \(\text{End}(U)\) of any indecomposable quadruple is described. One of the classification tools used in the paper is a bipartition matrix interpretation of quadruples, and the following useful lemma. If \(U\) is a quadruple without ``trivial'' direct summands and \(U'\) is the quadruple defined by setting \(U'_0=(U_1+U_2)\cap(U_3+U_4)\) and \(U'_j=U_j\cap U_0\), for \(j= 1,2,3,4\), then \(U\) is indecomposable if and only if \(U'\) is indecomposable.
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 2119124
zbMATH Keywords: quadruples
zbMATH Keywords: poset representations
zbMATH Keywords: regular representations
zbMATH Keywords: endomorphism algebras
MaRDI profile type: Publication
full work available at URL: j.laa.2004.05.010
OpenAlex ID: W2078618744
cites work: Q4318498
cites work: Endomorphism algebras of vector spaces with distinguished sets of subspaces
cites work: On four subspaces of a vector space
cites work: Q4095016
cites work: Q4046184
cites work: REPRESENTATION OF A TETRAD
cites work: Q4052279
cites work: Representation of partially ordered sets
DOI: 10.1016/J.LAA.2004.05.010
description: scientific article; zbMATH DE number 2119124