On the invariant faces associated with a cone-preserving map

For an $n\!\times \! n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of $\{ 1,\cdots ,n\}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\mathbb {R}^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If $A$ leaves invariant a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$ for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant join-irreducible faces of $K$, each containing in its relative interior a generalized eigenvector of $A$ corresponding to $\lambda$ (referred to as semi-distinguished $A$-invariant faces associated with $\lambda$), where $m_\lambda$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $\lambda$, but there is no such chain with more than $m_\lambda$ members. We introduce the important new concepts of semi-distinguished $A$-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding $n$ that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.