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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Crystal bases and simple modules for Hecke algebras of type $G(p,p,n)$
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by Jun Hu
Represent. Theory 11 (2007), 16-44
Published electronically: March 16, 2007


We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type $G(p,p,n)$. This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003), 7–20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004), 446–490]. Furthermore, we use Naito and Sagaki’s work [S. Naito & D. Sagaki, J. Algebra 251, (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to derive explicit formulas for the number of simple modules over these Hecke algebras. These formulas generalize earlier results of [M. Geck, Represent. Theory 4 (2000) 370-397] on the Hecke algebras of type $D_n$ (i.e., of type $G(2,2,n)$).
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Bibliographic Information
  • Jun Hu
  • Affiliation: Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
  • Email:
  • Received by editor(s): April 8, 2006
  • Published electronically: March 16, 2007
  • Additional Notes: This research was supported by the National Natural Science Foundation of China (Project 10401005) and by the Program for New Century Excellent Talents in University and partly by the URF of Victoria University of Wellington.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 11 (2007), 16-44
  • MSC (2000): Primary 20C08, 20C20, 17B37
  • DOI:
  • MathSciNet review: 2295687