Permutation invariant proper polyhedral cones and their Lyapunov rank

The Lyapunov rank of a proper cone K in a finite dimensional real Hilbert space is defined as the dimension of the space of all Lyapunov-like transformations on K, or equivalently, the dimension of the Lie algebra of the automorphism group of K. This (rank) measures the number of linearly independent bilinear relations needed to express a complementarity system on K (that arises, for example, from a linear program or a complementarity problem on the cone). Motivated by the problem of describing spectral/proper cones where the complementarity system can be expressed as a square system (that is, where the Lyapunov rank is greater than equal to the dimension of the ambient space), we consider proper polyhedral cones in Rn that are permutation invariant. For such cones we show that the Lyapunov rank is either 1 (in which case, the cone is irreducible) or n (in which case, the cone is isomorphic to R+n). In the latter case, we show that the corresponding spectral cone is isomorphic to a symmetric cone.