Crystal duality and Littlewood-Richardson rule of extremal weight crystals

We consider a category of \gl_\infty-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight \gl_\infty-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients

Littlewood identity and Crystal bases

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.Comment: 51 page

Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

The crystal base of the modified quantized enveloping algebras of type $A_{+\infty}$ or $A_\infty$ is realized as a set of integral bimatrices. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspondence.Comment: 30 page