Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

1 Hermitian eigenvalue problem For any n × n Hermitian matrix A, let λA = (λ1 ≥ · · · ≥ λn) be its set of eigenvalues written in descending order. (Recall that all the eigenvalues of a Hermitian matrix are real.) We recall the following classical problem. Problem 1. (The Hermitian eigenvalue problem) Given two n-tuples of nonincreasing real numbers: λ = (λ1 ≥ · · · ≥ λn) and μ = (μ1 ≥ · · · ≥ μn), determine all possible ν = (ν1 ≥ · · · ≥ νn) such that there exist Hermitian matrices A,B,C with λA = λ, λB = μ, λC = ν and C = A+B. Said imprecisely, the problem asks the possible eigenvalues of the sum of two Hermitian matrices with fixed eigenvalues. A conjectural solution of the above problem was given by Horn in 1962. For any positive integer r < n, inductively define the set S r as the set of triples (I, J,K) of subsets of [n] := {1, . . . , n} of cardinality r such that ∑ i∈I i+ ∑ j∈J j = r(r + 1)/2 + ∑ k∈K k (1)