Appears in collection : Combinatorics and Arithmetic for Physics

For the quiver Hecke algebra $R$, let $R$-gmod be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\mbox{-gmod}}$ be the localization of $R$-gmod, called ''localized quantum unipotent category``. It has been shown that the set of equivalence classes of simple objects up to grading shifts ${\rm Irr}(\widetilde{R\mbox{-gmod}})$ in $\widetilde{R\mbox{-gmod}}$ has a crystal structure, and ${\rm Irr}(\widetilde{R\mbox{-gmod}})$ is isomorphic to the so-called cellular crystal ${\mathbb B}_{\bf i}$ by M. Kashiwara and myself. This isomorphism induces a function $\varepsilon_i^*$ on ${\mathbb B}_{\bf i}$. We give an explicit form of $\varepsilon_i^*$, and using this, we give a characterization of the unit object of $\widetilde{R\mbox{-gmod}}$ for finite classical types, $A_n,\,B_n,\, C_n$ and $D_n$.
This is a joint work with Koh Matsuura.