Advertisement

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
Article

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

Keywords

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 

Notes

Acknowledgements

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

References

  1. 1.
    Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33, 839–867 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander, M., Reiten, I., Smalo, S.: Representation Theory of Artin Algebras, vol. 36. Cambridge studies in advanced mathematics, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, vol. 1. London Math. Soc. Student Texts 65, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bedard, R.: On commutation classes of reduced words in Weyl groups. Eur. J. Combin. 20, 483–505 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV–VI. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, (1968)Google Scholar
  6. 6.
    Brundan, J., Kleshchev, A., McNamara, P.J.: Homological properties of finite Khovanov–Lauda–Rouquier algebras. Duke Math. J. 163, 1353–1404 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chari, V.: Braid group actions and tensor products. Int. Math. Res. Not. 2002(7), 357–382 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chari, V., Pressley, A.: Yangians, integrable quantum systems and Dorey’s rule. Comm. Math. Phys. 181(2), 265–302 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Date, E., Okado, M.: Calculation of excitation spectra of the spin model related with the vector representation of the quantized affine algebra of type \(A^{(1)}\_n\). Int. J. Mod. Phys. A 9(3), 399–417 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frenkel, E., Hernandez, D.: Baxters relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hernandez, D.: Kirillov–Reshetikhin conjecture: the general case. Int. Math. Res. Not. 7, 149–193 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hernandez, D.: Simple tensor products. Invent. Math. 181, 649–675 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hernandez, D., Leclerc, B.: Quantum Grothendieck rings and derived Hall algebras. J. Reine Angew. Math. 701, 77–126 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kac, V.: Infinite dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kang, S.-J., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras. Invent. Math. 211(2), 591–685 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kang, S.-J., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II. Duke Math. J. 164(8), 1549–1602 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.: Simplicity of heads and socles of tensor products. Compos. Math. 151(2), 377–396 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV. Selecta Math. 22(4), 1987–2015 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. 69(2), 455–485 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kashiwara, M.: On level zero representations of quantum affine algebras. Duke. Math. J. 112, 117–175 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kato, S.: Poincaré–Birkhoff–Witt bases and Khovanov-Lauda-Rouquier algebras. Duke Math. J. 163(3), 619–663 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309–347 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363(5), 2685–2700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kleshchev, A., Ram, A.: Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words. Math. Ann. 349(4), 943–975 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Leclerc, B.: Imaginary vectors in the dual canonical basis of \(U\_q(\mathfrak{n})\). Transf. Groups 8(1), 95–104 (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1993)zbMATHGoogle Scholar
  28. 28.
    McNamara, P.: Finite dimensional representations of Khovanov-Lauda-Rouquier algebras I: finite type. J. Reine Angew. Math. 707, 103–124 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Oh, S.: Auslander-Reiten quiver of type A and generalized quantum affine Schur–Weyl duality. Trans. Am. Math. Soc. 369, 1895–1933 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Oh, S.: Auslander-Reiten quiver of type D and generalized quantum affine Schur–Weyl duality. J. Algebra 460, 203–252 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Oh, S.: The Denominators of normalized R-matrices of types \(A^{(2)}\_{2n-1}\), \(A^{(2)}\_{2n}\), \(B^{(1)}\_{n}\) and \(D^{(2)}\_{n+1}\). Publ. Res. Inst. Math. Sci. 51, 709–744 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Oh, S., Suh, U.: Combinatorial Auslander-Reiten quivers and reduced expressions. arXiv:1509.04820
  33. 33.
    Papi, P.: A characterization of a special ordering in a root system. Proc. Am. Math. 120, 661–665 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ringel, C.: PBW-bases of quantum groups. J. Reine Angew. Math. 470, 51–88 (1996)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Rouquier, R.: 2 Kac-Moody algebras. arXiv:0812.5023 (2008)
  36. 36.
    Rouquier, R.: Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19(2), 359–410 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Varagnolo, M., Vasserot, E.: Canonical bases and KLR algebras. J. Reine Angew. Math. 659, 67–100 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Young, C.A.S., Zegers, R.: Dorey’s rule and the q-characters of simply-laced quantum affine algebras. Comm. Math. Phys. 302(3), 789–813 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zelobenko, D.P.: Extremal cocycles on Weyl groups. Funktsional. Anal. i Prilozhen. 21(3), 11–21 (1987)MathSciNetGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsEwha Womans University SeoulSeoulKorea

Personalised recommendations