Inventiones mathematicae

, 178:451 | Cite as

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Article

Abstract

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

Mathematics Subject Classification (2000)

20C08 

References

  1. 1.
    Ariki, S.: On the decomposition numbers of the Hecke algebra of G(m,1,n). J. Math. Kyoto Univ. 36, 789–808 (1996) MathSciNetMATHGoogle Scholar
  2. 2.
    Ariki, S., Koike, K.: A Hecke algebra of (ℤ/rℤ) S n and construction of its irreducible representations. Adv. Math. 106, 216–243 (1994) CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Brundan, J.: Centers of degenerate cyclotomic Hecke algebras and parabolic category  \(\mathcal{O}\) . Represent. Theory 12, 236–259 (2008) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Brundan, J., Kleshchev, A.: The degenerate analogue of Ariki’s categorification theorem. arXiv:0901.0057
  5. 5.
    Brundan, J., Kleshchev, A.: Graded decomposition numbers for cyclotomic Hecke algebras. arXiv:0901.4450
  6. 6.
    Brundan, J., Kleshchev, A., Wang, W.: Graded Specht modules. arXiv:0901.0218
  7. 7.
    Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: category  \(\mathcal{O}\) . arXiv:0812.1090
  8. 8.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Basel (1997) MATHGoogle Scholar
  9. 9.
    Curtis, C., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Wiley, New York (1988) MATHGoogle Scholar
  10. 10.
    Dipper, R., James, G.D., Mathas, A.: Cyclotomic q-Schur algebras. Math. Z. 229, 385–416 (1998) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Fayers, M.: An extension of James’ conjecture. Int. Math. Res. Not. 10, 24 (2007) Google Scholar
  12. 12.
    Geck, M.: Representations of Hecke algebras at roots of unity. Astérisque 252, 33–55 (1998) MathSciNetGoogle Scholar
  13. 13.
    Grojnowski, I.: Affine \(\mathfrak{sl}_{p}\) controls the representation theory of the symmetric group and related Hecke algebras. arXiv:math.RT/9907129
  14. 14.
    Hoefsmit, P.: Representations of Hecke algebras of finite groups with BN-pairs of classical type. PhD thesis, University of British Columbia (1974) Google Scholar
  15. 15.
    Jucys, A.: Factorization of Young’s projection operators for symmetric groups. Litovsk. Fiz. Sb. 11, 1–10 (1971) MathSciNetGoogle Scholar
  16. 16.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. arXiv:0803.4121
  17. 17.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II. arXiv:0804.2080
  18. 18.
    Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge University Press, Cambridge (2005) MATHGoogle Scholar
  19. 19.
    Kleshchev, A.: Completely splittable representations of symmetric groups. J. Algebra 181, 584–592 (1996) CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Lauda, A.: Nilpotency in type A cyclotomic quotients. arXiv:0903.2992
  21. 21.
    Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2, 599–635 (1989) CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1993) MATHGoogle Scholar
  23. 23.
    Lyle, S., Mathas, A.: Blocks of cyclotomic Hecke algebras. Advances Math. 216, 854–878 (2007) CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Mathas, A.: Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group. University Lecture Series, vol. 15. American Mathematical Society, Providence (1999) MATHGoogle Scholar
  25. 25.
    Murphy, G.E.: The idempotents of the symmetric group and Nakayama’s conjecture. J. Algebra 81, 258–265 (1983) CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Ram, A.: Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. Lond. Math. Soc. 75, 99–133 (1997) CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Rogawski, J.D.: On modules over the Hecke algebra of a p-adic group. Invent. Math. 79, 443–465 (1985) CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Rouquier, R.: Derived equivalences and finite dimensional algebras. In: Proc. ICM (Madrid 2006), vol. 2, pp. 191–221. EMS Publishing House, Zürich (2006) Google Scholar
  29. 29.
    Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023
  30. 30.
    Ruff, O.: Completely splittable representations of symmetric groups and affine Hecke algebras. J. Algebra 305, 1197–1211 (2006) CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Turner, W.: Rock blocks. arXiv:0710.5462. Mem. Am. Math. Soc. (to appear)
  32. 32.
    Wenzl, H.: Hecke algebras of Type A n and subfactors. Invent. Math. 92, 349–383 (1988) CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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