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Convexity and Commuting Hamiltonians

Convexity and Commuting Hamiltonians M. F. ATIYAH §1. Introduction A well-known result of Schur [9] asserts that the diagonal elements (a ,..., a ) of l n annx n Hermitian matrix A satisfy a system of linear inequalities involving the eigenvalues (X ,..., X ). In geometric terms, regarding a and k as points in R" and i n allowing the symmetric group £„ to act by permutation of coordinates, this result takes the form (1.1) a is in the convex hull of the points Z A. The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie group G in the following manner. We consider the adjoint action of G on its Lie algebra L(G). If T is a maximal torus of G and W its Weyl group, then it is well known that W-orbits in L(T) correspond to G-orbits in L(G). Now fix a G-invariant metric on L(G), so that we can define orthogonal projection. Then Kostant's result isf (1.2) The orthogonal projection of a G-orbit onto L(T) coincides with the convex hull of the corresponding http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Convexity and Commuting Hamiltonians

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References (10)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/14.1.1
Publisher site
See Article on Publisher Site

Abstract

M. F. ATIYAH §1. Introduction A well-known result of Schur [9] asserts that the diagonal elements (a ,..., a ) of l n annx n Hermitian matrix A satisfy a system of linear inequalities involving the eigenvalues (X ,..., X ). In geometric terms, regarding a and k as points in R" and i n allowing the symmetric group £„ to act by permutation of coordinates, this result takes the form (1.1) a is in the convex hull of the points Z A. The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie group G in the following manner. We consider the adjoint action of G on its Lie algebra L(G). If T is a maximal torus of G and W its Weyl group, then it is well known that W-orbits in L(T) correspond to G-orbits in L(G). Now fix a G-invariant metric on L(G), so that we can define orthogonal projection. Then Kostant's result isf (1.2) The orthogonal projection of a G-orbit onto L(T) coincides with the convex hull of the corresponding

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1982

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