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

Hu, Jun

doi:10.1090/S1088-4165-07-00313-5

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

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

Published electronically: March 16, 2007

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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 $G(2,2,n)$).

References

Susumu Ariki, On the semi-simplicity of the Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr {\mathfrak {S}}_n$, J. Algebra 169 (1994), no. 1, 216–225. MR 1296590, DOI 10.1006/jabr.1994.1280
Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
Susumu Ariki, Cyclotomic $q$-Schur algebras as quotients of quantum algebras, J. Reine Angew. Math. 513 (1999), 53–69. MR 1713319, DOI 10.1515/crll.1999.063
Susumu Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $G(m,1,n)$ and Kleshchev multipartitions, Osaka J. Math. 38 (2001), no. 4, 827–837. MR 1864465
Susumu Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series, vol. 26, American Mathematical Society, Providence, RI, 2002. Translated from the 2000 Japanese edition and revised by the author. MR 1911030, DOI 10.1090/ulect/026
Susumu Ariki and Kazuhiko Koike, A Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr {\mathfrak {S}}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, DOI 10.1006/aima.1994.1057
Susumu Ariki and Andrew Mathas, The number of simple modules of the Hecke algebras of type $G(r,1,n)$, Math. Z. 233 (2000), no. 3, 601–623. MR 1750939, DOI 10.1007/s002090050489
Richard Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), no. 2, 501–512. MR 943273, DOI 10.1016/0021-8693(88)90275-X
Jonathan Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type $A$, Proc. London Math. Soc. (3) 77 (1998), no. 3, 551–581. MR 1643413, DOI 10.1112/S0024611598000562
Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$, Represent. Theory 5 (2001), 317–403. MR 1870595, DOI 10.1090/S1088-4165-01-00123-6
Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, DOI 10.1112/plms/s3-52.1.20
Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic $q$-Schur algebras, Math. Z. 229 (1998), no. 3, 385–416. MR 1658581, DOI 10.1007/PL00004665
Richard Dipper and Andrew Mathas, Morita equivalences of Ariki-Koike algebras, Math. Z. 240 (2002), no. 3, 579–610. MR 1924022, DOI 10.1007/s002090100371
Omar Foda, Bernard Leclerc, Masato Okado, Jean-Yves Thibon, and Trevor A. Welsh, Branching functions of $A^{(1)}_{n-1}$ and Jantzen-Seitz problem for Ariki-Koike algebras, Adv. Math. 141 (1999), no. 2, 322–365. MR 1671762, DOI 10.1006/aima.1998.1783
Jürgen Fuchs, Urmie Ray, and Christoph Schweigert, Some automorphisms of generalized Kac-Moody algebras, J. Algebra 191 (1997), no. 2, 518–540. MR 1448807, DOI 10.1006/jabr.1996.6907
Jürgen Fuchs, Bert Schellekens, and Christoph Schweigert, From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), no. 1, 39–97. MR 1403859
Meinolf Geck, On the representation theory of Iwahori-Hecke algebras of extended finite Weyl groups, Represent. Theory 4 (2000), 370–397. MR 1780716, DOI 10.1090/S1088-4165-00-00093-5
G. Genet and N. Jacon, Modular representations of cyclotomic Hecke algebras of type $G(r,p,n)$, preprint, math.RT/0409297.
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
Takahiro Hayashi, $q$-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), no. 1, 129–144. MR 1036118
Jun Hu, A Morita equivalence theorem for Hecke algebra ${\scr H}_q(D_n)$ when $n$ is even, Manuscripta Math. 108 (2002), no. 4, 409–430. MR 1923530, DOI 10.1007/s002290200272
Jun Hu, Crystal bases and simple modules for Hecke algebra of type $D_n$, J. Algebra 267 (2003), no. 1, 7–20. MR 1993465, DOI 10.1016/S0021-8693(03)00350-8
Jun Hu, Modular representations of Hecke algebras of type $G(p,p,n)$, J. Algebra 274 (2004), no. 2, 446–490. MR 2043358, DOI 10.1016/j.jalgebra.2003.10.029
J. Hu, Branching rules for Hecke Algebras of Type $D_{n}$, Math. Nachr. 280 (2007) 93–104.
J. Hu, The number of simple modules for the Hecke algebras of type G(r,p,n), preprint, math.RT/0601572.
N. Jacon, Représentations modulaires des algèbres de Hecke et des algèbres de Ariki-Koike, Ph.D. thesis, Université Claude Bernard Lyon I, 2004.
Nicolas Jacon, On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity, J. Math. Kyoto Univ. 44 (2004), no. 4, 729–767. MR 2118039, DOI 10.1215/kjm/1250281696
Michio Jimbo, Kailash C. Misra, Tetsuji Miwa, and Masato Okado, Combinatorics of representations of $U_q(\widehat {{\mathfrak {s}}{\mathfrak {l}}}(n))$ at $q=0$, Comm. Math. Phys. 136 (1991), no. 3, 543–566. MR 1099695
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
A. S. Kleshchev, Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. (2) 54 (1996), no. 1, 25–38. MR 1395065, DOI 10.1112/jlms/54.1.25
Peter Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3, 329–346. MR 1253196, DOI 10.1007/BF01231564
Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
Bernard Leclerc and Jean-Yves Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456. MR 1399410, DOI 10.1155/S1073792896000293
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Gunter Malle and Andrew Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra 205 (1998), no. 1, 275–293. MR 1631350, DOI 10.1006/jabr.1997.7339
Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak {s}\mathfrak {l}(n))$, Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
Satoshi Naito and Daisuke Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001), no. 1, 395–412. MR 1868198, DOI 10.1006/jabr.2001.8904
Satoshi Naito and Daisuke Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002), no. 1, 461–474. MR 1900295, DOI 10.1006/jabr.2001.9135
Christakis Pallikaros, Representations of Hecke algebras of type $D_n$, J. Algebra 169 (1994), no. 1, 20–48. MR 1296580, DOI 10.1006/jabr.1994.1270
Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no. 2, 267–297. MR 1722955, DOI 10.1215/S0012-7094-99-10010-X
M. Vazirani, Irreducible modules over the affine Hecke algebra: a strong multiplicity one result, Ph.D. thesis, University of California at Berkeley, 1998.