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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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
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 https://doi.org/10.1215/kjm/1250518452

[A$_2$]A2 S. Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $G(m,1,n)$ and Kleshchev multipartitions, to appear in Osaka J. Math..

Susumu Ariki and Kazuhiko Koike, A Hecke algebra of $({\bf Z}/r{\bf Z})\wr {\mathfrak S}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, DOI https://doi.org/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 https://doi.org/10.1007/s002090050489
George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, DOI https://doi.org/10.1016/0001-8708%2878%2990010-5
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak p}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172
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 https://doi.org/10.1112/S0024611598000562
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, DOI https://doi.org/10.1112/S0024611500012132

[BK$_2$]BK J. Brundan and A. Kleshchev, Projective representations of the symmetric group via Sergeev duality, to appear in Math. Z..

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 https://doi.org/10.1112/plms/s3-52.1.20

[G$_1$]G I. Grojnowski, Affine $\widehat {\mathfrak {sl}}_p$ controls the modular representation theory of the symmetric group and related Hecke algebras, preprint, 1999.

[G$_2$]G2 I. Grojnowski, Blocks of the cyclotomic Hecke algebra, preprint, 1999.

[GV]GV I. Grojnowski and M. Vazirani, Strong multiplicity one theorem for affine Hecke algebras of type $A$, Transf. Groups., to appear.

John F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. (2) 33 (1986), no. 3, 441–452. MR 850960, DOI https://doi.org/10.1112/jlms/s2-33.3.441
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, DOI https://doi.org/10.1112/plms/s3-65.3.475
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, DOI https://doi.org/10.1112/S002461159900177X
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219

[Kg]Kang S.-J. Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, preprint, Seoul National University, 2000.

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
Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9–36. MR 1458969, DOI https://doi.org/10.1215/S0012-7094-97-08902-X
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
A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI https://doi.org/10.1006/jabr.1995.1362
A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI https://doi.org/10.1006/jabr.1995.1362
Alexander Kleshchev, Branching rules for modular representations of symmetric groups. IV, J. Algebra 201 (1998), no. 2, 547–572. MR 1612335, DOI https://doi.org/10.1006/jabr.1997.7302
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, DOI https://doi.org/10.1112/S0024611597000427
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 https://doi.org/10.1155/S1073792896000293
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, DOI https://doi.org/10.1088/0305-4470/30/17/023

[Le]Leites D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1–64.

Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
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
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

[Mo]Morris A.O. Morris, The spin representations of the symmetric group, Canad. J. Math. 17 (1965), 543-549. involving shifted Young diagrams, Math. Proc. Camb. Phil. Soc. 99 (1986), 23-31.

Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, DOI https://doi.org/10.1006/aima.1997.1621
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, DOI https://doi.org/10.1007/PL00001384
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, DOI https://doi.org/10.1007/BF00402673
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, DOI https://doi.org/10.1155/S1073792800000234
Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. MR 1722115, DOI https://doi.org/10.1090/S1088-4165-99-00085-0
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 https://doi.org/10.1215/S0012-7094-99-10010-X

[VV]VV M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297.

[V$_1$]Vt M. Vazirani, Irreducible modules over the affine Hecke algebra: a strong multiplicity one result, Ph.D. thesis, UC Berkeley, 1999.

[V$_2$]V M. Vazirani, Filtrations on the Mackey decomposition for cyclotomic Hecke algebras, preprint, 1999.

A. V. Zelevinsky, Induced representations of reductive ${\mathfrak p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084
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