Littlewood identity and Crystal bases

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.