Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality

Abstract

We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey's rule for quantum affine algebras U-q(E-6,7,8((1))) and partial information of denominator formulas for U-q(E-6,7,8((1))). We also suggest conjecture on complete denominator formulas for U-q(E-6,7,8((1))).