Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Frobenius map for the centers of Hecke algebras


Authors: Jinkui Wan and Weiqiang Wang
Journal: Trans. Amer. Math. Soc. 367 (2015), 5507-5520
MSC (2010): Primary 20C08; Secondary 05E05
DOI: https://doi.org/10.1090/S0002-9947-2014-06211-9
Published electronically: December 18, 2014
MathSciNet review: 3347181
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [DJ] 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 (88m:20084), https://doi.org/10.1112/plms/s3-54.1.57
  • [FT] 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 (2000a:05207)
  • [Fra] Andrew Francis, The minimal basis for the centre of an Iwahori-Hecke algebra, J. Algebra 221 (1999), no. 1, 1-28. MR 1722901 (2000k:20005), https://doi.org/10.1006/jabr.1998.7925
  • [FrG] 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 (2008c:20005), https://doi.org/10.1016/j.jalgebra.2006.05.010
  • [FW] 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 (2010h:20011), https://doi.org/10.1090/conm/478/09317
  • [Fro] 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.
  • [GP] 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 (2002k:20017)
  • [GR] 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 (98c:20013)
  • [Jo] Lenny K. Jones, Centers of generic Hecke algebras, Trans. Amer. Math. Soc. 317 (1990), no. 1, 361-392. MR 948191 (90d:20030), https://doi.org/10.2307/2001467
  • [La] A.  Lascoux, The Hecke algebra and structure constants of the ring of symmetric polynomials, arXiv:math/0602379, 2006.
  • [Mac] 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 (96h:05207)
  • [Ram] Arun Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), no. 3, 461-488. MR 1134480 (93c:20029), https://doi.org/10.1007/BF01243921

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20C08, 05E05

Retrieve articles in all journals with MSC (2010): 20C08, 05E05


Additional Information

Jinkui Wan
Affiliation: School of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: wjk302@hotmail.com

Weiqiang Wang
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: ww9c@virginia.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06211-9
Keywords: Hecke algebras, centers, symmetric functions, Frobenius map
Received by editor(s): May 15, 2013
Received by editor(s) in revised form: June 6, 2013
Published electronically: December 18, 2014
Additional Notes: The first author was partially supported by NSFC-11101031, and the second author was partially supported by NSF DMS-1101268
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society