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].