Invariants of \(S_ 4\) and the shape of sets of vectors (Q1908903)

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.