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Kostant's convexity theorem
In mathematics, Kostant's convexity theorem, introduced by , states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of , and for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all ''n'' by ''n'' complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λ''n'') is the convex polytope with vertices all permutations of the coordinates of Λ. In fact this result is 'Kostant's linear convexity theorem'; the main focus of
is Kostant's nonlinear convexity theorem which involves the Iwasawa projection rather than the linear projection to the dual of a Cartan subalgebra.
Kostant used this to generalize the Golden–Thompson inequality to all compact groups.
==Compact Lie groups==
Let ''K'' be a connected compact Lie group with maximal torus ''T'' and Weyl group ''W'' = ''N''''K''(''T'')/''T''. Let their Lie algebras be and . Let ''P'' be the orthogonal projection of onto for some Ad-invariant inner product on . Then for ''X'' in , ''P''(Ad(''K'')⋅''X'') is the convex polytope with vertices ''w''(''X'') where ''w'' runs over the Weyl group.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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