Characterization of the unit object in localized quantum unipotent category

TL;DR

This paper characterizes the unit object in a localized quantum unipotent category by providing an explicit formula for a key crystal function, enhancing understanding of the categorical structure in classical finite types.

Contribution

It offers an explicit formula for the crystal function ps_i^* and characterizes the unit object in localized categories for classical finite types.

Findings

Explicit formula for ps_i^* function

Characterization of the unit object in localized categories

Application to classical finite types

Abstract

For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ in $\widetilde{R\hbox{-gmod}[w]}$ has a crystal structure, and $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ is isomorphic to the so-called cellular crystal $\mathbb B_{\mathbf i}$. This isomorphism induces a function $\varepsilon_i^*$ on $\mathbb B_{\mathbf i}$. We give an explicit formula of $\varepsilon_i^*$, and using this formula, we give a characterization of the unit object of $\widetilde{R\hbox{-gmod}[w]}$ for the case of classical finite types.