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

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.