Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 14 (2018), 007, 18 pages      arXiv:1706.04743      https://doi.org/10.3842/SIGMA.2018.007
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution

Olivier Dudas a and Nicolas Jacon b
a) Université Paris Diderot, UFR de Mathématiques, Bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France
b) Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Laboratoire de Mathématiques EA 4535, Moulin de la Housse BP 1039, 51100 Reims, France

Received June 17, 2017, in final form January 22, 2018; Published online January 30, 2018

Abstract
We study the effect of Alvis-Curtis duality on the unipotent representations of $\mathrm{GL}_n(q)$ in non-defining characteristic $\ell$. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both $\ell$ and the order of $q$ modulo $\ell$.

Key words: Mullineux involution; Alvis-Curtis duality; crystal graph; Harish-Chandra theory.

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References

  1. Alvis D., The duality operation in the character ring of a finite Chevalley group, hrefhttps://doi.org/10.1090/S0273-0979-1979-14690-1textitBull. Amer. Math. Soc. (N.S.) 1 (1979), 907-911.
  2. Bezrukavnikov R., Hilbert schemes and stable pairs, unpublished notes.
  3. Brundan J., Modular branching rules and the Mullineux map for Hecke algebras of type $A$, Proc. London Math. Soc. 77 (1998), 551-581.
  4. Cabanes M., Enguehard M., Representation theory of finite reductive groups, New Mathematical Monographs, Vol. 1, Cambridge University Press, Cambridge, 2004.
  5. Cabanes M., Rickard J., Alvis-Curtis duality as an equivalence of derived categories, in Modular Representation Theory of Finite Groups (Charlottesville, VA, 1998), de Gruyter, Berlin, 2001, 157-174.
  6. Chuang J., Rouquier R., Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification, Ann. of Math. 167 (2008), 245-298, math.RT/0407205.
  7. Chuang J., Rouquier R., Perverse equivalences, in preparation.
  8. Curtis C.W., Truncation and duality in the character ring of a finite group of Lie type, J. Algebra 62 (1980), 320-332.
  9. Curtis C.W., Reiner I., Methods of representation theory, Vol. II, With applications to finite groups and orders, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987.
  10. Deligne P., Lusztig G., Duality for representations of a reductive group over a finite field, J. Algebra 74 (1982), 284-291.
  11. Dipper R., Du J., Harish-Chandra vertices and Steinberg's tensor product theorems for finite general linear groups, Proc. London Math. Soc. 75 (1997), 559-599.
  12. Dreyfus-Schmidt L., Équivalences perverses splendides, conditions de stabilité et catégorification du complexe de Coxeter, Ph.D. Thesis, Paris Diderot - Paris 7, 2014.
  13. Dudas O., Varagnolo M., Vasserot E., Categorical actions on unipotent representations I. Finite unitary groups, arXiv:1509.03269.
  14. Ford B., Kleshchev A.S., A proof of the Mullineux conjecture, Math. Z. 226 (1997), 267-308.
  15. Geck M., On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017), 370-391.
  16. Geck M., Hiss G., Malle G., Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), 182-220.
  17. Geck M., Hiss G., Malle G., Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353-386.
  18. Gerber T., Hiss G., Jacon N., Harish-Chandra series in finite unitary groups and crystal graphs, Int. Math. Res. Not. 2015 (2015), 12206-12250, arXiv:1408.1210.
  19. Hiss G., Harish-Chandra series of Brauer characters in a finite group with a split $BN$-pair, J. London Math. Soc. 48 (1993), 219-228.
  20. Kleshchev A.S., Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. 54 (1996), 25-38.
  21. Lascoux A., Leclerc B., Thibon J.-Y., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205-263.
  22. Linckelmann M., Schroll S., A two-sided $q$-analogue of the Coxeter complex, J. Algebra 289 (2005), 128-134.
  23. Losev I., Supports of simple modules in cyclotomic Cherednik categories $\mathcal{O}$, arXiv:1509.00526.
  24. Losev I., Rational Cherednik algebras and categorification, in Categorification and Higher Representation Theory, Contemp. Math., Vol. 683, Amer. Math. Soc., Providence, RI, 2017, 1-40, arXiv:1509.08550.
  25. Misra K., Miwa T., Crystal base for the basic representation of $U_q(\widehat{\mathfrak{sl}}(n))$, Comm. Math. Phys. 134 (1990), 79-88.
  26. Mullineux G., Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups, J. London Math. Soc. 20 (1979), 60-66.
  27. Seeber J., On Harish-Chandra induction and the Hom-functor, MSc Thesis, RWTH Aachen University, 2016.

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