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

TL;DR

This paper explores the structure of Auslander-Reiten quivers in finite type ADE to gain insights into the representation theories of KLR-algebras, quantum groups, and quantum affine algebras, leading to new proofs and conjectures.

Contribution

It introduces new partial orders on positive roots and links their statistics to representation theories, providing proofs of Dorey's rule and conjectures for denominator formulas.

Findings

Proved Dorey's rule for $U_q(E_{6,7,8}^{(1)})$

Provided partial denominator formulas for $U_q(E_{6,7,8}^{(1)})$

Suggested conjectures on complete denominator formulas

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 $U_q(E_{6,7,8}^{(1)})$.