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

ABSTRACT

We begin with a review of combinatorial hives as introduced by Knutson and Tao, and focus on a conjecture by Danilov and Koshevoy for generating hives from Hermitian matrix pairs. 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 affirming computational results and opens a new area of research into the analysis of the random geometries of hive surfaces.

The second part of this article concerns methods of estimating Littlewood–Richardson coefficients from hives. We illustrate experimental confirmation and characterize two numerical algorithms: a rounded estimator on the continuous hive polytope following Narayanan, and a novel construction using a coordinate hit-and-run on the hive lattice.

KEYWORDS:

• combinatorial hives
• Littlewood-Richardson coefficients
• numerical optimization

2010 AMS Subject Classifcation:

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Acknowledgements

We would like to thank Hariharan Narayanan for the introduction to this construction and for many helpful conversations as an advisor and mentor. Additional thanks to Glenn Appleby in corresponding about [Appleby and Whitehead 14] and clarifying the history of hives from Hermitian matrix pairs with respect to reference [Danilov and Koshevoy 03]. We would also like to thank Github for a student developer grant and the hosting of private repositories. The algorithms discussed in this paper will be made freely available for public use and modification upon publication. This work was supported in part by the University of Washington.