Abacus-tournament models for Hall-Littlewood polynomials

摘要

In 2010, the first author introduced a combinatorial model for Schur polynomials based on labeled abaci. We generalize this construction to give analogous models for the Hall-Littlewood symmetric polynomials P-lambda, Q(lambda), and R-lambda, using objects called abacus-tournaments. We introduce various cancellation mechanisms on abacus-tournaments to obtain simpler combinatorial formulas and explain why these polynomials are divisible by certain products of t-factorials. These tools are then applied to give bijective proofs of several identities involving Hall-Littlewood polynomials, including the Pieri rule that expands the product P(mu)e(kappa) into a linear combination of Hall-Littlewood polynomials. (C) 2016 Elsevier B.V. All rights reserved.