Hives and Hermitian Matrices
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2023
Författare
Furufors, Karin
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
Hives, Hive constructions, Horn’s problem, the Saturation conjecture, Hermitian matrices, Littlewood-Richardson coefficients