Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case

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].

v3: 35 pages, introduction revised, Proposition 2.3(4) corrected, Proof of Proposition 2.3 is added, Remark 5.2 revised with references, Remark 5.6 added, Lemma 6.3 revised, Section 6.3 added, to appear in Letters in Mathematical Physics, v2: 29 pages, revised proof of Theorem 6.7, minor corrections; v1: 28 pages