Characterization of the unit object in localized quantum unipotent category
Koh Matsuura, Toshiki Nakashima
TL;DR
The paper characterizes the unit object in the localized localized quantum unipotent category for classical finite types and provides an explicit formula for the dual crystal statistic ε_i^* on the cellular crystal associated to a longest word. Building on the crystal structure on Irr( ilde{R-gmod}[w]) and the cellular crystal We show Irr( ilde{R-gmod}[w]) ≅ 𝔅_{oldsymbol i} and derive concrete ε_i^* expressions in types A_n, B_n, C_n, and D_n, enabling a precise criterion wt(X)=0 and ε_i^*(X)=0 for all i to characterize the unit object. The approach leverages localization via graded braiders, the induced crystal structure on the localized category, and braid-move isomorphisms to translate combinatorial data into module-theoretic interpretations. This yields explicit computational tools for understanding the unit in localized categories and deepens the link between quiver Hecke algebras, crystals, and cellular crystals in the classical finite-type setting.
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.