The Denominators of Normalized $R$-matrices of Types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ and $D_{n+1}^{(2)}$

Abstract

The denominators of normalized $R$-matrices provide important information on finite dimensional integrable representations over quantum affine algebras, and finite dimensional graded representations over quiver Hecke algebras by the generalized quantum affine Schur-Weyl duality functors. We compute the denominators of all normalized $R$-matrices between fundamental representations of types $A_{2n-1}^{(2)}$ $(n \ge 3)$, $A_{2n}^{(2)}$ $(n \ge 2)$, $B_{n}^{(1)}$ $(n \ge 3)$ and $D_{n+1}^{(2)}$ $(n \ge 2)$. Thus we can conclude that the normalized $R$-matrices of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ and $D_{3}^{(2)}$ have only simple poles, and of type $D_{n+1}^{(2)}$ $(n \ge 3)$ have double poles under certain conditions.