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

TL;DR

This paper computes the denominators of normalized R-matrices for fundamental representations across several quantum affine algebra types, revealing their pole structures and contributing to the understanding of their representation theory.

Contribution

It provides explicit calculations of denominators of normalized R-matrices for types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$, and $D_{n+1}^{(2)}$, clarifying their pole behaviors.

Findings

Normalized R-matrices of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ have only simple poles.

Type $D_{n+1}^{(2)}$ R-matrices have double poles under certain conditions.

Results aid in understanding finite-dimensional representations of quantum affine algebras.

Abstract

Denominators of normalized $R$-matrices provide important information on finite dimensional representations over quantum affine algebras, and 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 $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ and $D_{n+1}^{(2)}$. Thus we can conclude that the normalized $R$-matrices of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ have only simple poles, and of type $D_{n+1}^{(2)}$ have double poles under certain conditions.