GIT stability of linear maps on projective space with marked points

We construct moduli spaces of linear self-maps of PN with marked points, up to projective equivalence—that is, we let SLN+1 act simultaneously by conjugation on projective linear maps and diagonally on (PN)n, and we take the geometric invariant theory (GIT) quotient. These moduli spaces arise in algebraic dynamics and integrable systems. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps T with marked points. We show that GIT stability can be checked by counting the marked points on flags with certain Hessenberg functions relative to T. The proof is combinatorial: to describe the weight polytopes for this action, we compute the vertices and facets of certain convex polyhedra generated by roots of the AN lattice.