The symmetry of Littlewood-Richardson coefficients: a new hive model involutory bijection

Abstract

Littlewood--Richardson (LR) coefficients c^{\lambda}_{\mu\nu} may be evaluated by means of several combinatorial models. These include not only the original one, based on the LR rule for enumerating LR tableaux of skew shape \lambda /\mu and weight \nu, but also one based on the enumeration of LR hives with boundary edge labels \lambda, \mu, and \nu. Unfortunately, neither of these reveals in any obvious way the well-known symmetry property c^{\lambda}_{\mu\nu} = c^{\lambda}_{\nu\mu}. Here we introduce a map \sigma^(n) on LR hives that interchanges contributions to c^{\lambda}_{\mu\nu} and c^{\lambda}_{\nu\mu} for any partitions \lambda , \mu, \nu of lengths no greater than n, and then we prove that it is a bijection, thereby making manifest the required symmetry property. The map \sigma^(n) involves repeated path removals from a given LR hive with boundary edge labels (\lambda,\mu,\nu) that give rise to a sequence of hives whose left-hand boundary edge labels define a partner LR hive with boundary edge labels (\lambda,\nu,\mu). A new feature of our hive model is its realization in terms of edge labels and rhombus gradients, with the latter playing a key role in defining the action of path removal operators in a manner designed to preserve the required hive conditions. A consideration of the detailed properties of the path removal procedures also leads to a wholly combinatorial selfcontained hive based proof that \sigma^(n) is an involution.