Master's Thesis presentation, Karin Furufors

Abstract:

Given two Hermitian matrices, M and N, what can be said about the eigenvalues of their sum L = M + N? In 1962 mathematician Alfred Horn conjectured that a recursive system of inequalities would give both necessary and sufficient conditions to characterize the eigenvalues of the sum in terms of the eigenvalues of the summands. The conjecture was proven just before the turn of the millennium, with important contributions by A. Knutson and T. Tao. Knutson and Tao also introduced a combinatorial object called a hive, which provides a more tractable expression for Horn's inequalities. The hive is a tiling of small equilateral triangles inside a larger equilateral triangle, with the interior vertices being nodes to be assigned values. The eigenvalues of the triple define the boundaries of the larger triangle and constraints are given by the rhombus inequalities. From Knutson and Tao’s proof, we know that a Hermitian matrix triple (M, N, L = M + N) with given integral eigenspectra exists if and only if an integral labelling of the hive nodes can be found. A question that remains is how to construct a hive given a triple of Hermitian matrices. A proposal for a hive construction was given in 2014 by G. Appleby and T. Whitehead, but some flaws have since been pointed out. In this thesis we take their proposal for a hive construction, translate it for the special case of diagonal matrices and prove algebraically that this new formulation generates hives for simultaneously diagonalizable Hermitian matrices generally.

Opponent: Ludwig Scherqvist Halldner