Littlewood identity and Crystal bases

TL;DR

This paper introduces a new combinatorial tableau model for crystal bases of classical Lie algebra modules, extending Littlewood-Richardson rules and connecting to Lie superalgebra characters through superization.

Contribution

It provides a novel tableau-based combinatorial model for crystals of type B and C modules, extending classical rules and linking to superalgebra characters via super duality.

Findings

New tableau model for crystal bases of types B and C

Extended Littlewood-Richardson rule for these modules

Superization of tableau model producing superalgebra characters

Abstract

We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ in terms of classical Young tableux. We then obtain a new description of its Littlewood-Richardson rule and a maximal Levi branching rule in terms of classical Littlewood-Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks. We also show that this tableau model admits a natural superization and it produces the characters of irreducible highest weight modules over orthosymplectic Lie superalgebras, which correspond to the integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ under the Cheng-Lam-Wang's super duality.