On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called moments characteristic polynomials random Haar-distributed unitary and symplectic matrices, as matrix size N goes to infinity. These quantities depend on two parameters k q when both them are positive integers it shown that these fact N. In this paper we classify integer roots moreover prove themselves satisfy a symmetry property. This confirms some predictions from thesis Bailey [7]. The proof uses Ehrhart-Macdonald reciprocity for rational convex polytopes bijections between lattice points polytopes.