ANGULAR MOMENTUM AND HORN'S PROBLEM

We prove a conjecture made by the first named author: Given an n-body central configuration X-0 in the euclidean space E of dimension 2p, let Im F be the set of decreasing real p-tuples (v(1), v(2),..., v(p)) such that {+/- v(1), +/- v(2),..., +/- v(p)} is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of X-0 in E. Then Im F is a convex polytope. The proof consists in showing that there exist two, generically (p-1)-dimensional, convex polytopes P-1 and P-2 in such that P-1 subset of Im F subset of P-2 and that these two polytopes coincide. P-1, introduced earlier in a paper by the first author, is the set of spectra corresponding to the hermitian structures J on E which are "adapted" to the symmetries of the inertia matrix S-0; it is associated with Horn's problem for the sum of p x p 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 asociated with Horii's problem for the sum of 2p x 2p real symmetric matrices having each the same spectrum as S-0. The equality P-1 = P-2 follows directly from a deep combinatorial lemma by S. Fomin, W. Fulton, C. K. Li, and Y. T. Coon, which characterizes those of the sums of two 2p x 2p real symmetric matrices with the same spectrum which are hermitian for some hermitian structure.