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 2 n + 1 , D n + 1 , E 6 and D 4 associated to (triply) twisted Coxeter elements. Using the quivers of type A 2 n + 1 and D n + 1 , we describe the denominator formulas and Dorey's rule for quantum affine algebras U q ′ ( B n + 1 ( 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 n + 1 ( 1 ) ) (resp. U q ′ ( C n ( 1 ) ) ) using certain statistics on any folded AR-quiver of type A 2 n + 1 (resp. D n + 1 ) and Dorey's rule for U q ′ ( B n + 1 ( 1 ) ) (resp. U q ′ ( C n ( 1 ) ) ) 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 4 ( 1 ) ) and U q ′ ( G 2 ( 1 ) ) .