Title: Honey from the Hives: A Theoretical and Computational Exploration of Combinatorial Hives,
Abstract: In the first half of this manuscript, we begin with a brief review of
combinatorial hives as introduced by Knutson and Tao, and focus on a conjecture
by Danilov and Koshevoy for generating such a hive from Hermitian matrix pairs
through an optimization scheme. We examine a proposal by Appleby and Whitehead
in the spirit of this conjecture and analytically elucidate an obstruction in
their construction for guaranteeing hive generation, while detailing stronger
conditions under which we can produce hives with almost certain probability. We
provide the first mapping of this prescription onto a practical algorithmic
space that enables us to produce affirming computational results and open a new
area of research into the analysis of the random geometries and curvatures of
hive surfaces from select matrix ensembles.
The second part of this manuscript concerns Littlewood-Richardson
coefficients and methods of estimating them from the hive construction. We
illustrate experimental confirmation of two numerical algorithms that we
provide as tools for the community: one as a rounded estimator on the
continuous hive polytope volume following a proposal by Narayanan, and the
other as a novel construction using a coordinate hit-and-run on the hive
lattice itself. We compare the advantages of each, and include numerical
results on their accuracies for some tested cases.