Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case

TL;DR

This paper extends the understanding of prefundamental modules over twisted quantum loop algebras, providing character formulas and realizing minuscule modules in terms of unipotent quantum coordinate rings for specific types.

Contribution

It introduces a character formula for twisted prefundamental modules and realizes minuscule modules via unipotent quantum coordinate rings for types A_{2n-1}^{(2)} and D_{n+1}^{(2)}.

Findings

Derived character formulas from untwisted cases using folding maps.

Realized minuscule prefundamental modules in terms of quantum coordinate rings.

Extended previous realizations to twisted quantum loop algebra types.

Abstract

We study the prefundamental modules $L_{s,a}^{\pm}$ over the Borel subalgebras of the twisted quantum loop algebras, which are introduced by Wang. A character formula for $L_{s,a}^{\pm}$ is obtained from that for the prefundamental modules over the untwisted quantum loop algebras by applying a character folding map. This allows us to realize minuscule prefundamental modules $L_{s,a}^{\pm}$ for types $A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$ in terms of the unipotent quantum coordinate ring associated with the $s$-th level $0$ fundamental weight, where $s = 1$ for type $A_{2n-1}^{(2)}$ and $s = n$ for type $D_{n+1}^{(2)}$. This result is a continuation of the realization of (co)minuscule prefundamental modules established by earlier works [J-Kwon-Park, Int. Math. Res. Not., 2023] and [J-Kwon-Park, J. Algebra, 2025].