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

Abstract3or any N × N symmetric real matrix W, the subspaces of RN which are invariant under πTWπ for all N × N permutation matrices π are characterized. For most W, the only such invariant subspaces are RN and {O}, but there are many exceptions. The related problem of characterizing the class of linear transformations that leave a given space and all its permutations invariant is also solved