Unipotent quantum coordinate ring and cominuscule prefundamental representations

TL;DR

This paper explores the realization of prefundamental modules in unipotent quantum coordinate rings, establishing character equalities and module structures, especially for cominuscule cases, advancing understanding in quantum algebra representations.

Contribution

It demonstrates the character equality of prefundamental modules and unipotent quantum coordinate rings, and constructs module structures for cominuscule cases, extending prior work in quantum algebra.

Findings

Character of $L_{r,a}^{\u00b1}$ equals that of $U_q^-(w_r)$.

Existence of a $U_q(\u2208b)$-module structure on $U_q^-(w_r)$ for cominuscule $r$.

Isomorphism between this module and $L_{r,a\u03b7_r}^$.

Abstract

We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to $L_{r,a\eta_r}^\pm$ for some $\eta_r \in \mathbb{C}^\times$.