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Representation Theory
Representation Theory
ISSN 1088-4165

 

Crystal bases and simple modules for Hecke algebras of type $ G(p,p,n)$


Author: Jun Hu
Journal: Represent. Theory 11 (2007), 16-44
MSC (2000): Primary 20C08, 20C20, 17B37
Published electronically: March 16, 2007
MathSciNet review: 2295687
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Abstract: 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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Additional Information

Jun Hu
Affiliation: Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: junhu303@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S1088-4165-07-00313-5
PII: S 1088-4165(07)00313-5
Keywords: Cyclotomic Hecke algebra, Fock space, crystal basis, Kleshchev multipartition, Lakshmibai--Seshadri path.
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.