Crystal base of the negative half of quantum orthosymplectic superalgebra

TL;DR

This paper constructs a crystal base for the negative part of a quantum orthosymplectic superalgebra, providing a combinatorial and crystal-theoretic framework that extends understanding of superalgebra representations.

Contribution

It introduces a new crystal base construction for the negative half of quantum orthosymplectic superalgebras and describes embeddings via combinatorial and PBW basis methods.

Findings

Constructed crystal base as a limit of q-deformed oscillator representations

Provided a combinatorial description of embeddings between crystals

Introduced a new crystal realization of Burge correspondence for orthosymplectic type

Abstract

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding.