Frobenius map for the centers of Hecke algebras

Wan, Jinkui; Wang, Weiqiang

doi:10.1090/S0002-9947-2014-06211-9

Frobenius map for the centers of Hecke algebras
HTML articles powered by AMS MathViewer

by Jinkui Wan and Weiqiang Wang PDF

Trans. Amer. Math. Soc. 367 (2015), 5507-5520 Request permission

Abstract:

We introduce a commutative associative graded algebra structure on the direct sum $\mathcal Z$ of the centers of the Hecke algebras associated to the symmetric groups in $n$ letters for all $n$. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from $\mathcal Z$ to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.

References

Richard Dipper and Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57–82. MR 872250, DOI 10.1112/plms/s3-54.1.57
S. Fomin and J.-Y. Thibon, Inverting the Frobenius map, Algebra i Analiz 10 (1998), no. 3, 183–192; English transl., St. Petersburg Math. J. 10 (1999), no. 3, 545–552. MR 1628046
Andrew Francis, The minimal basis for the centre of an Iwahori-Hecke algebra, J. Algebra 221 (1999), no. 1, 1–28. MR 1722901, DOI 10.1006/jabr.1998.7925
Andrew R. Francis and John J. Graham, Centres of Hecke algebras: the Dipper-James conjecture, J. Algebra 306 (2006), no. 1, 244–267. MR 2271582, DOI 10.1016/j.jalgebra.2006.05.010
Andrew Francis and Weiqiang Wang, The centers of Iwahori-Hecke algebras are filtered, Representation theory, Contemp. Math., vol. 478, Amer. Math. Soc., Providence, RI, 2009, pp. 29–37. MR 2513264, DOI 10.1090/conm/478/09317
F. Frobenius, Über die Charactere der symmetrischen Gruppe, Sitzungberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1900), 516–534; reprinted in Gessamelte Abhandlungen 3, 148–166.
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Meinolf Geck and Raphaël Rouquier, Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 251–272. MR 1429875
Lenny K. Jones, Centers of generic Hecke algebras, Trans. Amer. Math. Soc. 317 (1990), no. 1, 361–392. MR 948191, DOI 10.1090/S0002-9947-1990-0948191-6
A. Lascoux, The Hecke algebra and structure constants of the ring of symmetric polynomials, arXiv:math/0602379, 2006.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Arun Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), no. 3, 461–488. MR 1134480, DOI 10.1007/BF01243921