Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 499–554 | Cite as

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

  • Se-jin OhEmail author


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


Auslander–Reiten quiver Positive roots Convex orders [Q]-distance [Q]-socle KLR algebra Generalized KLR-type Schur–Weyl duality Distance polynomial Exceptional E-types 

Mathematics Subject Classification

Primary 05E10 16T30 17B37 Secondary 81R50 



The author would like to express his sincere gratitude to Professor Masaki Kashiwara, Myungho Kim and Chul-hee Lee for many fruitful discussions.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsEwha Womans University SeoulSeoulKorea

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