A super Littlewood--Richardson type rule
Nohra Hage
TL;DR
This work develops a superalgebraic analogue of the Littlewood--Richardson rule for super Schur functions on signed alphabets, anchored by a super Robinson--Schensted correspondence. It provides a concrete combinatorial interpretation of the super LR coefficients $c_{\lambda,\mu}^{\nu}$ as the number of Littlewood--Richardson tableaux of shape $\nu/\mu$ and weight $\lambda$, derived from pairs of super tableaux via a robust insertion framework. The main identities $S_\lambda S_\mu = \sum_\nu c_{\lambda,\mu}^{\nu} S_\nu$ and $S_{\nu/\lambda} = \sum_\mu c_{\lambda,\mu}^{\nu} S_\mu$ are established in the super plactic setting for arbitrary orderings of $\mathcal{S}$, unifying classical signed-alphabet results and Kashiwara-crystal perspectives. The framework generalizes prior work, recovers the non-signed case and links to representation theory and mathematical physics via superalgebras and super tableaux.
Abstract
We introduce a super version of the Littlewood--Richardson rule for super Schur functions over signed alphabets. We give in particular combinatorial interpretations of the super Littlewood--Richardson coefficients using the properties of super Young tableaux, which have found rich applications in representation theory, algebraic combinatorics, and mathematical physics.