Honeycombs and Sums of Hermitian Matrices

Horn’s conjecture [Ho], which given the spectra of two Hermitianmatrices describes the possible spectra of the sum, was recently settled in the affirmative. We discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a honeycomb; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by “classical vs. quantum” analogies. IfH is an n×nHermitian matrix, then the spectrum λ = (λ1 ≥ . . . ≥ λn) can be written as a weakly decreasing sequence of n real numbers. Conversely, for every spectrum λ we can form the set Oλ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). If λ, μ, ν are three spectra, we define the relation λ⊞ μ ∼c ν (1) if there exist Hermitian matrices Hλ ∈ Oλ, Hμ ∈ Oμ, Hν ∈ Oν such that Hλ + Hμ = Hν. (The “c” in ∼c stands for “classical”; we will define a quantum analogue ∼q later on.) For instance, one can easily verify that (3) ⊞ (4) ∼c (7), (3, 0)⊞ (4, 0) ∼c (7, 0), (3, 0)⊞ (4, 0) ∼c (4, 3), (2, 0)⊞ (2, 0) ∼c (3, 1) but that (3) ⊞ (4) 6∼c (5), (3, 0) ⊞ (4, 0) 6∼c (8,−1) In 1912, Hermann Weyl [W] posed the problem of determining the set of triples λ, μ, ν for which (1) held. Or more informally: given the eigenvalues of two Hermitian matrices Hλ and Hμ, what are all the possible eigenvalues of the sum Hλ + Hμ? The purpose of this article is to describe the successful resolution to this problem, based on recent breakthroughs [Kl, HR, KT, KTW]. It is fairly easy to obtain necessary conditions in order for (1) to hold. For instance, from the simple observation that the trace of Hλ + Hμ must equal the sum of the traces of Hλ and Hμ, we obtain the condition ν1 + . . . + νn = λ1 + . . .+ λn + μ1 + . . . + μn. (2) Another immediate constraint is that ν1 ≤ λ1 + μ1, (3) since the largest eigenvalue ofHλ+Hμ is at most the sum ofHλ andHμ’s individual largest eigenvalues. (Exercise for the reader: show equality occurs exactly when the same vector Date: February 1, 2008. AK’s research was partially conducted for the Clay Mathematics Institute. TT is supported by the Clay Mathematics Institute and by grants from the Sloan and Packard foundations.