# Candidate for the crystal $B(-\infty)$ for the queer Lie superalgebra

## TL;DR

This paper constructs a crystal model for the queer Lie superalgebra's polynomial representations as a direct limit of semistandard decomposition tableaux, providing a combinatorial framework and recovering representations from this limit.

## Contribution

It introduces a new crystal model for the queer Lie superalgebra's polynomial representations as a direct limit of tableaux, extending combinatorial descriptions.

## Key findings

- Existence of the direct limit crystal model for the queer Lie superalgebra.
- Extension of the tableau model to describe the limit combinatorially.
- Polynomial representations can be recovered from the limit in most cases.

## Abstract

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.