Honeycombs for 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].