Angular momentum and Horn's problem

Abstract

We prove a conjecture made by the first author: given an n-body central configuration X_0 in the euclidean space R^{2p}, let Im F be the set of ordered real p-tuples {\nu_1,\nu_2,...,\nu_p} such that {\pm i\nu_1,\pm i\nu_2,...,\pm i\nu_p} is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of X_0 in R^{2p}. Then Im F is a convex polytope. The proof consists in showing that there exist two (p-1)-dimensional convex polytopes P_1 and P_2 in R^{p} such that Im F lies between P_1 and P_2 and that these two polytopes coincide. Introduced in \cite{C1}, P_1 is the set of spectra corresponding to the hermitian structures J on R^{2p} which are "adapted" to the symmetries of the inertia matrix S_0; it is associated with Horn's problem for the sum of pxp real symmetric matrices with spectra sigma_- and sigma_+ whose union is the spectrum of S_0; P_2 is the orthogonal projection onto the set of "hermitian spectra" of the polytope P associated with Horn's problem for the sum of 2px2p real symmetric matrices having each the same spectrum as S_0. The equality P_1=P_2 follows directly from a deep combinatorial lemma, proved by Fomin, Fulton, Li and Poon, which implies that among the sums of two 2px2p real symmetric matrices A and B with the same spectrum, those C=A+B which are hermitian for some hermitian structure play a central role.