Hives and Hermitian Matrices

Abstract

Given two Hermitian matrices, M and N, what can be said about the eigenvalues of their sum L = M + N? In 1962, A. Horn conjectured that a set of recursive inequalities would fully characterize the eigenspectrum of the sum in terms of the eigenspectra of the summands. Just before the turn of the millennium the saturation conjecture was proven, and as a consequence, Horn’s conjecture was established. In their proof, A. Knutson and T. Tao introduced a combinatorial object, known as a hive, that rephrases Horn’s inequalities into more tractable expressions. The path from Hermitian matrices to hives, however, remains largely unexplored. A proposal for a hive construction, that is, a mapping from matrices to hives, was put forth by G. Appleby and T.Whitehead in 2014, but the proof of the construction’s validity has since come under question. In this thesis, the hive construction proposal by Appleby and Whitehead serves as a basis for a reformulated hive construction, adapted to the restricted setting of pairs of simultaneously diagonalizable Hermitian matrices. Equipped with this modified formulation, we proceed to prove algebraically that the construction indeed generates hives for simultaneously diagonalizable Hermitian matrices generally, thereby providing a mapping from matrix triples to hives under these special circumstances.