Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

TL;DR

This paper provides a combinatorial framework for understanding crystal bases of quantum queer superalgebras using semistandard decomposition tableaux, including insertion algorithms and tensor product decompositions.

Contribution

It introduces explicit combinatorial models and algorithms for the crystal bases of U_q(q(n)) modules, advancing the understanding of their structure.

Findings

Explicit combinatorial realization of crystal B(λ)

Insertion scheme for semistandard decomposition tableaux

Algorithms for tensor product decomposition

Abstract

In this paper, we give an explicit combinatorial realization of the crystal B(\lambda) for an irreducible highest weight U_q(q(n))-module V(\lambda) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.