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

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 Uq(E6,7,8(1)) and partial information of denominator formulas for Uq(E6,7,8(1)). We also suggest conjecture on complete denominator formulas for Uq(E6,7,8(1)).