Twisted Coxeter elements and Folded AR-quivers via Dynkin diagram automorphisms:II

TL;DR

This paper provides a combinatorial interpretation of Dorey's rule for type C_n using twisted AR-quivers of type D_{n+1}, linking Coxeter elements, adapted classes, and Dynkin quivers.

Contribution

It introduces twisted Dynkin quivers of type D_{n+1} and establishes one-to-one correspondences with twisted Coxeter elements and AR-quivers, expanding the combinatorial framework.

Findings

Combinatorial interpretation of Dorey's rule for type C_n.

Identification of twisted AR-quivers as a cluster point.

Establishment of bijections between twisted Coxeter elements, classes, and quivers.

Abstract

As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. Combinatorial properties of twisted AR-quivers are useful to understand not only Dorey's rule but also other notions in the representation theory of the quantum affine algebra $U_q'(C_n^{(1)})$ such as denominator formulas. In addition, unlike twisted adapted classes of type $A_{2n-1}$ in the previous paper, we show twisted AR-quivers of type $D_{n+1}$ consist of the cluster point called twisted adapted cluster point. Hence, by introducing new combinatorial objects called twisted Dynkin quivers of type $D_{n+1}$, we give one to one correspondences between twisted Coxeter elements, twisted adapted classes and twisted AR-quivers.