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

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 $U'q(G^{(1)}{2})$.