A categorification of $\mathfrak{q}(2)$-crystals

TL;DR

This paper develops a categorification of rak{q}(2)-crystals within the singular rak{gl}_n-category al O_n, extending previous rak{gl}_2-crystal structures and exploring their properties.

Contribution

It introduces a new categorification framework for rak{q}(2)-crystals on al O_n, expanding the understanding of crystal structures in representation theory.

Findings

Extended rak{gl}_2-crystal structure to rak{q}(2)-crystals

Analyzed properties of the rak{q}(2)-crystal al O_n

Connected categorification with existing crystal structures

Abstract

We provide a categorification of $\mathfrak{q}(2)$-crystals on the singular $\mathfrak{gl}_{n}$-category ${\mathcal O}_{n}$. Our result extends the $\mathfrak{gl}_{2}$-crystal structure on ${\rm Irr} ({\mathcal O}_{n})$ defined by Bernstein-Frenkel-Khovanov. Further properties of the ${\mathfrak q}(2)$-crystal ${\rm Irr}({\mathcal O}_{n})$ are also discussed.