Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

TL;DR

This paper introduces twisted and folded Auslander-Reiten quivers for specific types and applies them to derive denominator formulas and Dorey's rule for quantum affine algebras, advancing understanding in their representation theory.

Contribution

It develops new twisted and folded AR-quivers for types A, D, E, and D4, and uses them to describe key formulas and rules in quantum affine algebra representation theory.

Findings

Derived denominator formulas for U'_q(B^{(1)}_{n+1}) and U'_q(C^{(1)}_{n})

Applied folded AR-quivers to Dorey's rule

Proposed conjectural formulas for F^{(1)}_{4} and G^{(1)}_{2}

Abstract

In this paper, we introduce twisted and folded AR-quivers of type $A_{2n+1}$, $D_{n+1}$, $E_6$ and $D_4$ associated to (triply) twisted Coxeter elements. Using the quivers of type $A_{2n+1}$ and $D_{n+1}$, we describe the denominator formulas and Dorey's rule for quantum affine algebras $U'_q(B^{(1)}_{n+1})$ and $U'_q(C^{(1)}_{n})$, which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for $U'_q(B^{(1)}_{n+1})$ (resp. $U'_q(C^{(1)}_{n})$) using certain statistics on any folded AR-quiver of type $A_{2n+1}$ (resp. $D_{n+1}$) and Dorey's rule for $U'_q(B^{(1)}_{n+1})$ (resp. $U'_q(C^{(1)}_{n})$) applying the notion of minimal pairs in a twisted AR-quiver. By adopting the same arguments, we propose the conjectural denominator formulas and Dorey's rule for $U'_q(F^{(1)}_{4})$ and $U'_q(G^{(1)}_{2})$.