Honeycombs for Hall polynomials

TL;DR

This paper introduces a honeycomb-based formulation of Hall polynomials, proving key algebraic properties and establishing their equivalence with traditional definitions through linear algebraic methods.

Contribution

It presents a novel honeycomb formulation for Hall polynomials, extending combinatorial techniques used for Littlewood--Richardson coefficients.

Findings

Proved a Pieri rule for honeycomb Hall polynomials

Established associativity of the honeycomb formula

Demonstrated equality with classical Hall polynomials

Abstract

We propose a new formulation of Hall polynomials in terms of honeycombs, which were previously introduced in the context of the Littlewood--Richardson rule. We prove a Pieri rule and associativity for our honeycomb formula, thus showing equality with Hall polynomials. Our proofs are linear algebraic in nature, extending nontrivially the corresponding bijective results for ordinary Littlewood--Richardson coefficients [A. Knutson, T. Tao, C. Woodward, 2004].