J an 2 00 0 EIGENVALUES , INVARIANT FACTORS , HIGHEST WEIGHTS , AND SCHUBERT CALCULUS
نویسنده
چکیده
We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.
منابع مشابه
m at h . A G ] 2 A ug 1 99 9 EIGENVALUES , INVARIANT FACTORS , HIGHEST WEIGHTS , AND SCHUBERT CALCULUS
We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.
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The set of possible spectra ( ; ; ) of zero-sum triples of Hermitian matrices forms a polyhedral cone [H], whose facets have been already studied in [Kl, HR, T, Be] in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities d...
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We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Recent breakthroughs, prima...
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The set of possible spectra (λ, μ, ν) of zero-sum triples of Hermitian matrices forms a polyhedral cone [H], whose facets have been already studied in [Kl, HR, T, Be] in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities...
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Recently Klyachko [K] has given linear inequalities on triples (λ, μ, ν) of dominant weights of GLn(C) necessary for the the corresponding Littlewood-Richardson coefficient dim(Vλ⊗Vμ⊗Vν) GLn(C) to be positive. We show that these conditions (and an evident congruency condition) are also sufficient, which was known as the saturation conjecture. In particular this proves Horn’s conjecture, giving ...
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