Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case
Il-Seung Jang
TL;DR
This work extends the theory of prefundamental representations to twisted quantum affine algebras by folding untwisted data through a Grothendieck ring map π. It proves that, for A_{2n-1}^{(2)} and D_{n+1}^{(2)}, minuscule prefundamental modules L_{s,a}^{±} can be realized as modules on the unipotent quantum coordinate ring U_q^-(w_s) at level zero, up to a spectral parameter shift, and gives explicit character and ell-weight formulas via a folding of the untwisted data. The results build on and extend previous co/minuscule constructions, providing concrete algebraic realizations and character formulas that align twisted prefundamental modules with unipotent coordinate rings. The findings contribute to a deeper understanding of the representation theory of twisted quantum loop algebras and have implications for Q-systems and Bethe Ansatz via transparent module realizations.
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].