Honeycombs for Hall polynomials (Q2185213)

Let \(P^{\lambda}\) denote the Hall-Littlewood symmetric function indexed the partition \(\lambda\). Hall polynomials appear as the structure constants \(c^{\lambda,\mu}_{\nu}\), the coefficients of \(P^{\nu}\) in the product of \(P^{\lambda}P^{\mu}\), namely \[P^{\lambda}P^{\mu}=\sum c^{\lambda,\mu}_{\nu} P^{\nu}.\] The main result of this paper is to give a new combinatorial interpretation of \(c^{\lambda,\mu}_{\nu}\) by using weighted honeycombs with boundaries \(\lambda\), \(\mu\), \(\nu\), which recovers a formulation of the Littewood-Richardson rule for Schur polynomials. Though the proof of the main result follows the spirit of \textit{A. Knutson} et al. [Electron. J. Comb. 11, No. 1, Research paper R61, 18 p. (2004; Zbl 1053.05119)], it is highly nontrivial, which takes more than ten pages to prove the associativity.