Remote Access Representation Theory
Green Open Access

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
DOI: https://doi.org/10.1090/S1088-4165-07-00313-5
Published electronically: March 16, 2007
MathSciNet review: 2295687
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$).


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

  • 1. S. Ariki, On the semi-simplicity of the Hecke algebra of $ (Z/rZ)\wr {\mathfrak{S}}_n$, J. Alg. 169 (1994) 216-225. MR 1296590 (95h:16020)
  • 2. S. Ariki, On the decomposition numbers of the Hecke algebra of $ G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996) 789-808. MR 1443748 (98h:20012)
  • 3. S. Ariki, Cyclotomic $ q$-Schur algebras as quotients of quantum algebras, J. Reine Angew. Math. 513 (1999) 53-69. MR 1713319 (2001a:16065)
  • 4. S. Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $ G(m,1,n)$ and Kleshchev multi-partitions, Osaka J. Math. (4) 38 (2001) 827-837. MR 1864465 (2002i:20004)
  • 5. S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, Translated from the 2000 Japanese edition and revised by the author, University Lecture Series, 26, American Mathematical Society, Providence, RI, 2002. MR 1911030 (2004b:17022)
  • 6. S. Ariki and K. Koike, A Hecke algebra of $ (\mathbb{Z}/r\mathbb{Z})\wr\mathfrak{S}_n$ and construction of its representations, Adv. Math. 106 (1994) 216-243. MR 1279219 (95h:20006)
  • 7. S. Ariki and A. Mathas, The number of simple modules of the Hecke algebras of type $ G(r,1,n)$, Math. Z. (3) 233 (2000) 601-623. MR 1750939 (2001e:20007)
  • 8. R.E. Borcherds, Generalized Kac-Moody algebras, J. Alg. (2) 115 (1988) 501-512. MR 943273 (89g:17004)
  • 9. J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type $ A$, Proc. London. Math. Soc. (3) 77 (1998) 551-581. MR 1643413 (2000d:20007)
  • 10. J. Brundan and A. Kleshchev, Hecke-Clifford superalgebras, crystals of type $ A_{2l}^{(2)}$ and modular branching rules for $ \widehat{S}_n$, Representation Theory 5 (2001) 317-403. MR 1870595 (2002j:17024)
  • 11. M. Broué and G. Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993) 119-189. MR 1235834 (94m:20095)
  • 12. C. W. Curtis and L. Reiner, Methods of Representations Theory, I, Wiley-Interscience, New York, 1981. MR 0632548 (82i:20001)
  • 13. R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London. Math. Soc. (3) 52 (1986) 20-52. MR 812444 (88b:20065)
  • 14. R. Dipper, G.D. James and A. Mathas, Cyclotomic $ q$-Schur algebras, Math. Z. 229 (1998) 385-416. MR 1658581 (2000a:20033)
  • 15. R. Dipper and A. Mathas, Morita equivalence of Ariki-Koike algebras, Math. Z. 240 (2002) 579-610. MR 1924022 (2003h:20011)
  • 16. O. Foda, B. Leclerc, M. Okado, J.-Y. Thibon and T. Welsh, Branching functions of $ A_{(n-1)}^{(1)}$ and Jantzen-Seitz problem for Ariki-Koike algebras, Adv. Math. (2) 141 (1999) 322-365. MR 1671762 (2000f:17036)
  • 17. J. Fuchs, U. Ray and C. Schweigert, Some automorphisms of generalized Kac-Moody algebras, J. Alg. 191 (1997) 518-540. MR 1448807 (98g:17025)
  • 18. J. Fuchs, B. Schellekens and C. Schweigert, From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996) 39-97. MR 1403859 (98i:17021)
  • 19. M. Geck, On the representation theory of Iwahori-Hecke algebras of extended finite Weyl groups, Represent. Theory 4 (2000) 370-397. MR 1780716 (2002d:20006)
  • 20. G. Genet and N. Jacon, Modular representations of cyclotomic Hecke algebras of type $ G(r,p,n)$, preprint, math.RT/0409297.
  • 21. J.J. Graham and G.I. Lehrer, Cellular algebras Invent. Math. 123 (1996) 1-34. MR 1376244 (97h:20016)
  • 22. T. Hayashi, $ q$-analogues of Clifford and Weyl algebras--spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990) 129-144. MR 1036118 (91a:17015)
  • 23. J. Hu, A Morita equivalence theorem for Hecke algebras of type $ D_n$ when $ n$ is even, Manuscr. Math. 108 (2002) 409-430. MR 1923530 (2003h:20012)
  • 24. J. Hu, Crystal basis and simple modules for Hecke algebra of type $ D_n$, J. Alg. (1) 267 (2003) 7-20. MR 1993465 (2004d:20005)
  • 25. J. Hu, Modular representations of Hecke algebras of type $ G(p,p,n)$, J. Alg. (2) 274 (2004) 446-490. MR 2043358 (2004m:20014)
  • 26. J. Hu, Branching rules for Hecke Algebras of Type $ D_{n}$, Math. Nachr. 280 (2007) 93-104.
  • 27. J. Hu, The number of simple modules for the Hecke algebras of type G(r,p,n), preprint, math.RT/0601572.
  • 28. N. Jacon, Représentations modulaires des algèbres de Hecke et des algèbres de Ariki-Koike, Ph.D. thesis, Université Claude Bernard Lyon I, 2004.
  • 29. N. Jacon, On the parameterization of the simple modules for Ariki-Koike algebras at roots of unity, J. Math. Kyoto Univ. 44 (2004) 729-767. MR 2118039 (2006c:20008)
  • 30. M. Jimbo, K. Misra, T. Miwa and M. Okado, Combinatorics of representations of $ U_q(\hat{sl}(n))$ at $ q=0$, Comm. Math. Phys. 136 (1991) 543-566. MR 1099695 (93a:17015)
  • 31. M. Kashiwara, On crystal bases of the $ q$-analogue of universal enveloping algebras, Duke J. Math. 63 (1991) 465-516. MR 1115118 (93b:17045)
  • 32. A. Kleshchev, Branching rules for the modular representations of symmetric groups III; some corollaries and a problem of Mullineux, J. London Math. Soc. 54 (1995) 25-38. MR 1395065 (96m:20019c)
  • 33. P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994) 329-346. MR 1253196 (95f:17023)
  • 34. P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995) 499-525. MR 1356780 (96m:17011)
  • 35. A. Lascoux, B. Leclerc and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases quantum affine algebras, Comm. Math. Phys. 181 (1996) 205-263. MR 1410572 (97k:17019)
  • 36. B. Leclerc and J.-Y. Thibon, Canonical bases of $ q$-deformed Fock spaces, Int. Math. Res. Notices, 9 (1996) 447-456. MR 1399410 (97h:17023)
  • 37. G. Lusztig, Introduction to Quantum Groups, Progress in Math. 110 Birkhäuser, Boston, 1990. MR 1227098 (94m:17016)
  • 38. G. Malle and A. Mathas, Symmetric cyclotomic Hecke algebras, J. Alg. 205 (1998) 275-293. MR 1631350 (99g:20026)
  • 39. T. Misra and K.C. Miwa, Crystal bases for the basic representations of $ U_q(\ksl_n)$, Comm. Math. Phys. 134 (1990) 79-88. MR 1079801 (91j:17021)
  • 40. S. Naito and D. Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism, J. Alg. 245 (2001) 395-412. MR 1868198 (2002j:17026)
  • 41. S. Naito and D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Alg. 251 (2002) 461-474. MR 1900295 (2003d:17031)
  • 42. C. Pallikaros, Representations of Hecke algebras of type $ D_n$, J. Alg. 169 (1994) 20-48. MR 1296580 (95k:20015)
  • 43. M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebras, Duke Math. J. 100 (1999) 267-297. MR 1722955 (2001c:17029)
  • 44. M. Vazirani, Irreducible modules over the affine Hecke algebra: a strong multiplicity one result, Ph.D. thesis, University of California at Berkeley, 1998.

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20C08, 20C20, 17B37

Retrieve articles in all journals with MSC (2000): 20C08, 20C20, 17B37


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: https://doi.org/10.1090/S1088-4165-07-00313-5
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.

American Mathematical Society