Honey from the Hives: A Theoretical and Computational Exploration of Combinatorial Hives

摘要

In the first half of this manuscript, we begin with a brief review ofcombinatorial hives as introduced by Knutson and Tao, and focus on a conjectureby Danilov and Koshevoy for generating such a hive from Hermitian matrix pairsthrough an optimization scheme. We examine a proposal by Appleby and Whiteheadin the spirit of this conjecture and analytically elucidate an obstruction intheir construction for guaranteeing hive generation, while detailing strongerconditions under which we can produce hives with almost certain probability. Weprovide the first mapping of this prescription onto a practical algorithmicspace that enables us to produce affirming computational results and open a newarea of research into the analysis of the random geometries and curvatures ofhive surfaces from select matrix ensembles. The second part of this manuscript concerns Littlewood-Richardsoncoefficients and methods of estimating them from the hive construction. Weillustrate experimental confirmation of two numerical algorithms that weprovide as tools for the community: one as a rounded estimator on thecontinuous hive polytope volume following a proposal by Narayanan, and theother as a novel construction using a coordinate hit-and-run on the hivelattice itself. We compare the advantages of each, and include numericalresults on their accuracies for some tested cases.