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

TL;DR

This paper explores combinatorial hives through a blend of theoretical analysis and computational methods, focusing on matrix-based generation, algorithmic implementation, and estimation of Littlewood-Richardson coefficients.

Contribution

It introduces new conditions for hive generation from matrices, maps these to practical algorithms, and develops two numerical methods for estimating Littlewood-Richardson coefficients.

Findings

Identified obstructions in existing hive generation proposals.

Developed algorithms for generating hives from matrix pairs.

Validated two numerical algorithms for estimating Littlewood-Richardson coefficients.

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.