The involutive nature of the Littlewood-Richardson commutativity bijection

Abstract

Littlewood-Richardson (LR) coefficients $c_{\mu\nu}^\lambda$ may be evaluated by means of several combinatorial models, including the original LR tableaux of skew shape $\lambda/\mu$ and weight $\nu$ and the LR hives with boundary edge labels $\lambda$, $\mu$ and $\nu$. Unfortunately, neither of these reveal in any obvious way the well-known symmetry property $c_{\mu\nu}^\lambda=c_{\nu\mu}^\lambda$. Here we introduce two maps, $\rho^{(n)}$ on LR tableaux and $\sigma^{(n)}$ on LR hives, that each interchange contributions to $c_{\mu\nu}^\lambda$ and $c_{\nu\mu}^\lambda$ for any partitions $\lambda$, $\mu$, $\nu$ of lengths no greater than $n$, and then prove not only that each of them is a bijection, thereby making manifest the required symmetry property, but also that both maps are involutions. The map $\rho^{(n)}$ involves the iterative action of deletion operators on a given LR tableau of skew shape $\lambda/\mu$ and weight $\nu$, that produce a sequence of successively smaller tableaux whose consecutive inner shapes define a certain Gelfand-Tsetlin pattern and determine a partner LR tableau of skew shape $\lambda/\nu$ and weight $\mu$. Similarly, 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 the same Gelfand-Tsetlin pattern as before, which is sufficient to determine a partner LR hive with boundary edge labels $(\lambda,\nu,\mu)$. The deletions in tableaux are organised so as to preserve the semistandard and lattice permutation properties of LR tableaux, while the path removals in hives are designed to preserve both the triangle condition on edge labels and the hive rhombus gradient positivity conditions. At all stages illustrative examples are provided.