Title: Permutation invariant proper polyhedral cones and their Lyapunov rank,
Abstract: 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 $\mathbb{R}^n$ 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 the nonnegative orthart in $\mathbb{R}^n$). In the latter case,
we show that the corresponding spectral cone is isomorphic to a symmetric cone.