Characterization of invariant spaces under a symmetric real matrix and its permutations (Q923655)

The authors solve the following two closely related problems: (1) given is a real symmetric matrix W of order n; find the subspaces of \(R^ n\) invariant for all the matrices \(\pi^ TW\pi\), where \(\pi\) runs over all n! permutation matrices; (2) given is a subspace \(V\subset R^ n\); find all real symmetric matrices W such that V would be invariant for all matrices \(\pi^ TW\pi.\) For most of W the problem (1) has only trivial solutions \(\{\) \(0\}\) and \(\{R^ n\}\). The authors find all matrices W for which there exist nontrivial solutions of the problem (1) and give an explicit description of the set \(\{\) \(W\}\) for any nontrivial V in the problem (2).