Advertisement

Mathematische Zeitschrift

, Volume 293, Issue 3–4, pp 935–955 | Cite as

An analogue of row removal for diagrammatic Cherednik algebras

  • Chris Bowman
  • Liron SpeyerEmail author
Article

Abstract

We prove an analogue of James–Donkin row removal theorems for diagrammatic Cherednik algebras. This is one of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic. As a special case, our results yield a new reduction theorem for graded decomposition numbers and extension groups for cyclotomic q-Schur algebras.

Notes

Acknowledgements

The authors would like to thank the Royal Commission for the Exhibition of 1851 and the Japan Society for the Promotion of Science for their financial support.

References

  1. 1.
    Bowman, C., Cox, A., Speyer, L.: A family of graded decomposition numbers for diagrammatic Cherednik algebras. Int. Math. Res. Not. IMRN 2017(9), 2686–2734 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brundan, J., Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras. Invent. Math. 178(3), 451–484 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brundan, J., Kleshchev, A.: Graded decomposition numbers for cyclotomic Hecke algebras. Adv. Math. 222(6), 1883–1942 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bowman, C.: The many graded cellular bases of Hecke algebras, arXiv:1702.06579v5, preprint (2017)
  5. 5.
    Bonnafé, C., Rouquier, R.: Cellules de Calogero–Moser, arXiv:1302.2720, preprint (2013)
  6. 6.
    Bonnafé, C., Rouquier, R.: Calogero–Moser versus Kazhdan–Lusztig cells. Pacific J. Math. 261(1), 45–51 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bowman, C., Speyer, L.: Kleshchev’s decomposition numbers for diagrammatic Cherednik algebras. Trans. Am. Math. Soc. 370(5), 3551–3590 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chlouveraki, M., Gordon, I., Griffeth, S.: Cell modules and canonical basic sets for Hecke algebras from Cherednik algebras. New Trends in Noncommutative Algebra 562, 77–89 (2012). (Contemp. Math. Amer. Math. Soc, Providence, RI)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chlouveraki, M., Jacon, N.: Schur elements for the Ariki–Koike algebra and applications. J. Algebraic Combin. 35(2), 291–311 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chuang, J., Miyachi, H., Tan, K.M.: Row and column removal in the \(q\)-deformed Fock space. J. Algebra 254(1), 84–91 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dipper, R., James, G.D., Mathas, A.: Cyclotomic \(q\)-Schur algebras. Math. Z. 229(3), 385–416 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Donkin, S.: A note on decomposition numbers for general linear groups and symmetric groups. Math. Proc. Camb. Philos. Soc. 97(1), 57–62 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Donkin, S.: The \(q\)-Schur Algebra. London Mathematical Society Lecture Note Series, vol. 253. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  14. 14.
    Donkin, S.: Tilting modules for Algebraic Groups and Finite Dimensional Algebras. Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332. Cambridge University Press, Cambridge, pp. 215–257 (2007)Google Scholar
  15. 15.
    Elias, B., Losev, I.: Modular representation theory in type \(A\) via Soergel bimodules, arXiv:1701.00560, preprint (2017)
  16. 16.
    Fayers, M., Lyle, S.: Row and column removal theorems for homomorphisms between Specht modules. J. Pure Appl. Algebra 185, 147–164 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fayers, M., Speyer, L.: Generalised column removal for graded homomorphisms between Specht modules. J. Algebraic Combin. 44(2), 393–432 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Geck, M.: Kazhdan–Lusztig cells and decomposition numbers. Represent. Theory 2, 264–277 (1998). (electronic)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Geck, M., Jacon, N.: Representations of Hecke Algebras at Roots of Unity, Algebra and Applications, vol. 15. Springer, London (2011)zbMATHGoogle Scholar
  20. 20.
    Geck, M., Rouquier, R.: Filtrations on projective modules for Iwahori-Hecke algebras, Modular representation theory of finite groups (Charlottesville, VA, 1998), de Gruyter, Berlin, pp. 211–221 (2001)Google Scholar
  21. 21.
    Jacon, N.: Canonical basic sets for Hecke algebras, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., vol. 392, Amer. Math. Soc., Providence, RI, pp. 33–41 (2005)Google Scholar
  22. 22.
    James, G.D.: On the decomposition matrices of the symmetric groups, III. J. Algebra 71(1), 115–122 (1981)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kang, S.-J., Kashiwara, M.: Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras. Invent. Math. 190(3), 699–742 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lyle, S., Mathas, A.: Row and column removal theorems for homomorphisms of Specht modules and Weyl modules. J. Algebraic Combin. 22, 151–179 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Losev, I.: Proof of Varagnolo–Vasserot conjecture on cyclotomic categories \(\cal{O}\). Sel. Math. 22(2), 631–668 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mathas, A.: Cyclotomic quiver Hecke algebras of type \(A\), Modular representation theory of finite and \(p\)-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 30, World Sci. Publ., Hackensack, NJ, pp. 165–266 (2015)Google Scholar
  27. 27.
    Rouquier, R., Shan, P., Varagnolo, M., Vasserot, E.: Categorifications and cyclotomic rational double affine Hecke algebras. Invent. Math. 204(3), 671–786 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Webster, B.: Weighted Khovanov–Lauda–Rouquier algebras, arXiv:1209.2463, preprint (2012)
  29. 29.
    Webster, B.: Knot invariants and higher representation theory. Mem. Am. Math. Soc. 250(1191), 133 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Webster, B.: Rouquier’s conjecture and diagrammatic algebra. Forum Math. Sigma 5, e27, 71 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Williamson, G.: Schubert calculus and torsion explosion. J. Am. Math. Soc. 30(4), 1023–1046 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.University of KentCanterburyUK
  2. 2.University of VirginiaCharlottesvilleUSA

Personalised recommendations