Unipotent quantum coordinate ring and cominuscule prefundamental representations
Il-Seung Jang, Jae-Hoon Kwon, Euiyong Park
TL;DR
This work extends the realization of prefundamental modules $L_{r,a}^{\pm}$ within unipotent quantum coordinate rings to all cominuscule indices in untwisted affine types, showing the ordinary character of $L_{r,a}^{\pm}$ coincides with that of $U_q^-(w_r)$ and constructing a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$ that realizes $L_{r,a\eta_r}^{\pm}$ up to a spectral-shift parameter $\eta_r$ in cominuscule cases. The authors develop a unified framework based on braid group symmetries of the $q$-boson algebra, extended affine Weyl group translations, and quantum shuffle technology to identify root vectors and compute ell-weights, culminating in explicit character formulas and realizations for cominuscule negative and positive prefundamental modules. These results generalize prior cominuscule/minuscule cases and link unipotent quantum coordinate rings to the category $\mathcal{O}$ for $U_q(\mathfrak{b})$, offering new avenues for understanding prefundamental representations and their spectral parameter shifts. The work advances the understanding of the interplay between quantum unipotent structures and prefundamental modules in affine settings, with potential implications for related $Q\widetilde{Q}$-systems and representation theory of quantum groups.
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 $L_{r,aη_r}^\pm$ for some $η_r \in \mathbb{C}^\times$.