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Article overview
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Angular momentum and Horn's problem | | Alain Chenciner
; Hugo Jimenez Perez
; | | Date: |
23 Oct 2011 | | 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 {
u_1,
u_2,...,
u_p} such that {pm i
u_1,pm i
u_2,...,pm
i
u_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. | | Source: | arXiv, 1110.5030 | | Services: | Forum | Review | PDF | Favorites | |
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