Hecke-Clifford superalgebras, crystals of type 𝐴_{2ℓ}⁽²⁾ and modular branching rules for ̂𝑆_{𝑛}

Brundan, Jonathan; Kleshchev, Alexander

doi:10.1090/S1088-4165-01-00123-6

Hecke-Clifford superalgebras, crystals of type $A_{2\ell}^{(2)}$ and modular branching rules for $\widehat{S}_n$

Authors: Jonathan Brundan and Alexander Kleshchev
Journal: Represent. Theory 5 (2001), 317-403
MSC (2000): Primary 17B67, 20C08, 20C20, 17B10, 17B37
DOI: https://doi.org/10.1090/S1088-4165-01-00123-6
Published electronically: October 24, 2001
MathSciNet review: 1870595
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Abstract: This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.

[AF] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. MR 0417223
[A] 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, https://doi.org/10.1215/kjm/1250518452
[A] S. Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type and Kleshchev multipartitions, to appear in Osaka J. Math..
[AK] Susumu Ariki and Kazuhiko Koike, A Hecke algebra of (𝑍/𝑟𝑍)≀𝔖_{𝔫} and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, https://doi.org/10.1006/aima.1994.1057
[AM] 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, https://doi.org/10.1007/s002090050489
[Be] George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, https://doi.org/10.1016/0001-8708(78)90010-5
[BZ] I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive 𝔭-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172
[B] 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, https://doi.org/10.1112/S0024611598000562
[BK] Jonathan Brundan and Alexander Kleshchev, On translation functors for general linear and symmetric groups, Proc. London Math. Soc. (3) 80 (2000), no. 1, 75–106. MR 1719176, https://doi.org/10.1112/S0024611500012132
[BK] J. Brundan and A. Kleshchev, Projective representations of the symmetric group via Sergeev duality, to appear in Math. Z..
[DJ] 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, https://doi.org/10.1112/plms/s3-52.1.20
[G] I. Grojnowski, Affine $\widehat{\mathfrak{sl}}_p$ controls the modular representation theory of the symmetric group and related Hecke algebras, preprint, 1999.
[G] I. Grojnowski, Blocks of the cyclotomic Hecke algebra, preprint, 1999.
[GV] I. Grojnowski and M. Vazirani, Strong multiplicity one theorem for affine Hecke algebras of type , Transf. Groups., to appear.
[H] John F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. (2) 33 (1986), no. 3, 441–452. MR 850960, https://doi.org/10.1112/jlms/s2-33.3.441
[JS] Jens C. Jantzen and Gary M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. (3) 65 (1992), no. 3, 475–504. MR 1182100, https://doi.org/10.1112/plms/s3-65.3.475
[JN] A. R. Jones and M. L. Nazarov, Affine Sergeev algebra and 𝑞-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. (3) 78 (1999), no. 3, 481–512. MR 1674836, https://doi.org/10.1112/S002461159900177X
[Kc] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
[Kg] S.-J. Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, preprint, Seoul National University, 2000.
[Ka] Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
[KS] Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9–36. MR 1458969, https://doi.org/10.1215/S0012-7094-97-08902-X
[Kt] Shin-ichi Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 929–943 (1982). MR 656065
[K] A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, https://doi.org/10.1006/jabr.1995.1362
[K] A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, https://doi.org/10.1006/jabr.1995.1362
[K] Alexander Kleshchev, Branching rules for modular representations of symmetric groups. IV, J. Algebra 201 (1998), no. 2, 547–572. MR 1612335, https://doi.org/10.1006/jabr.1997.7302
[K] Alexander Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. (3) 75 (1997), no. 3, 497–558. MR 1466660, https://doi.org/10.1112/S0024611597000427
[LLT] 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
[LT] Bernard Leclerc and Jean-Yves Thibon, Canonical bases of 𝑞-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456. MR 1399410, https://doi.org/10.1155/S1073792896000293
[LT] Bernard Leclerc and Jean-Yves Thibon, 𝑞-deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A 30 (1997), no. 17, 6163–6176. MR 1482704, https://doi.org/10.1088/0305-4470/30/17/023
[Le] D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1-64.
[ML] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
[Man] Yuri I. Manin, Gauge field theory and complex geometry, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1997. Translated from the 1984 Russian original by N. Koblitz and J. R. King; With an appendix by Sergei Merkulov. MR 1632008
[MM] Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of 𝑈_{𝑞}(𝔰𝔩(𝔫)), Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
[Mo] A. O. Morris, The spin representation of the symmetric group, Canadian J. Math. 17 (1965), 543–549. MR 175964, https://doi.org/10.4153/CJM-1965-055-0
[N] Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, https://doi.org/10.1006/aima.1997.1621
[OV] Andrei Okounkov and Anatoly Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), no. 4, 581–605. MR 1443185, https://doi.org/10.1007/PL00001384
[O] G. I. Olshanski, Quantized universal enveloping superalgebra of type 𝑄 and a super-extension of the Hecke algebra, Lett. Math. Phys. 24 (1992), no. 2, 93–102. MR 1163061, https://doi.org/10.1007/BF00402673
[Sch] Olivier Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 8 (2000), 413–440. MR 1753691, https://doi.org/10.1155/S1073792800000234
[S] Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. MR 1722115, https://doi.org/10.1090/S1088-4165-99-00085-0
[S] 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, https://doi.org/10.1215/S0012-7094-99-10010-X
[VV] M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267-297.
[V] M. Vazirani, Irreducible modules over the affine Hecke algebra: a strong multiplicity one result, Ph.D. thesis, UC Berkeley, 1999.
[V] M. Vazirani, Filtrations on the Mackey decomposition for cyclotomic Hecke algebras, preprint, 1999.
[Z] A. V. Zelevinsky, Induced representations of reductive 𝔭-adic groups. II. On irreducible representations of 𝔊𝔏(𝔫), Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084
[Z] Andrey V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR 643482