Eigenvalues 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
Eigenvalues of Products of Unitary Matrices and Quantum Schubert Calculus
We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured two-sphere with prescribed holonomies, and the decomposition of fusion product of representations of SU(n), in the large level limit. In the second part of the paper we investigate how various aspe...
متن کاملm 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.
متن کاملJ an 2 00 0 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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Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A1, . . . , Ar ∈ Cn×n with the spectrum of the sum A1 + · · ·+Ar. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.
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Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schuber...
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