Invariants of \(S_ 4\) and the shape of sets of vectors
type: Item
instance of: scholarly article
title: Invariants of \(S_ 4\) and the shape of sets of vectors
zbMATH Open document ID: 0849.20007
DOI: 10.1007/BF01613616
author: Helmer Aslaksen
author: Shih-Ping Chan
author: Tor H. Gulliksen
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1996-03-07T00:00:00Z
review text: Let \(\mathcal M\) be the vector space of symmetric \(n \times n\) matrices over \(\mathbb{R}\) with zeros on the diagonal. A representation \(\phi\) of \(S_n\) of degree \(n \choose 2\) is defined by \(\phi(P)M = P^t AP\), where \(M\in {\mathcal M}\) and \(P \in S_n\) is identified as the corresponding \(n \times n\) permutation matrix. The basic problem considered is to determine the invariants of this representation. For \(n = 3\), the authors point out that the answer is easy and for \(n = 4\), they are able to give an explicit solution which is rather complicated. In addition, using computer calculations, they have been able to obtain all the basic syzygies but these are too complicated to be stated. They add some comments to show how impossible the situation becomes even for \(n = 5\). They also carefully explain the relevance of this problem to `shapes' of sets of vectors.
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 852853
zbMATH Keywords: symmetric matrices
zbMATH Keywords: permutation matrix
zbMATH Keywords: invariants
zbMATH Keywords: explicit solution
zbMATH Keywords: computer calculations
zbMATH Keywords: basic syzygies
describes a project that uses: Macaulay2
describes a project that uses: Maple
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 852853