Hecke-Clifford superalgebras, crystals of type and modular branching rules for

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

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.

**[AF]**F. Anderson and K. Fuller,*Rings and categories of modules*, Springer-Verlag, 1974. MR**54:5281****[A]**S. Ariki, On the decomposition numbers of the Hecke algebra of type ,*J. Math. Kyoto Univ.***36**(1996), 789-808. MR**98h:20012****[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]**S. Ariki and K. Koike, A Hecke algebra of and construction of its irreducible representations,*Advances Math.***106**(1994), 216-243. MR**95h:20006****[AM]**S. Ariki and A. Mathas, The number of simple modules of the Hecke algebras of type ,*Math. Z.***233**(2000), 601-623. MR**2001e:20007****[Be]**G. Bergman, The diamond lemma for ring theory,*Advances Math.***29**(1978), 178-218. MR**81b:16001****[BZ]**I. Bernstein and A. Zelevinsky, Induced representations of reductive -adic groups, I,*Ann. Sci. Ecole Norm. Sup.***10**(1977), 441-472. MR**58:28310****[B]**J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type ,*Proc. London Math. Soc.***77**(1998), 551-581. MR**2000d:20007****[BK]**J. Brundan and A. Kleshchev, Translation functors for general linear and symmetric groups,*Proc. London Math. Soc.***80**(2000), 75-106. MR**2000j:20080****[BK]**J. Brundan and A. Kleshchev, Projective representations of the symmetric group via Sergeev duality, to appear in*Math. Z.*.**[DJ]**R. Dipper and G. James, Representations of Hecke algebras of general linear groups.*Proc. London Math. Soc.***52**(1986), 20-52. MR**88b:20065****[G]**I. Grojnowski, Affine 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]**J. F. Humphreys, Blocks of projective representations of the symmetric groups,*J. London Math. Soc.***33**(1986), 441-452. MR**87k:20027****[JS]**J. C. Jantzen and G. M. Seitz, On the representation theory of the symmetric groups,*Proc. London Math. Soc.***65**(1992), 475-504. MR**93k:20026****[JN]**A. Jones and M. Nazarov, Affine Sergeev algebra and -analogues of the Young symmetrizers for projective representations of the symmetric group,*Proc. London Math. Soc.***78**(1999), 481-512. MR**2000a:20021****[Kc]**V. Kac,*Infinite-dimensional Lie algebras*, Cambridge University Press, third edition, 1990. MR**92k:17038****[Kg]**S.-J. Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, preprint, Seoul National University, 2000.**[Ka]**M. Kashiwara, ``On crystal bases'', in: Representations of groups (Banff 1994),*CMS Conf. Proc.***16**(1995), 155-197. MR**97a:17016****[KS]**M. Kashiwara and Y. Saito, Geometric construction of crystal bases,*Duke Math. J.***89**(1997), 9-36. MR**99e:17025****[Kt]**S. Kato, Irreducibility of principal series representations for Hecke algebras of affine type,*J. Fac. Sci. Univ. Tokyo Sect. IA Math.***28**(1981), 929-943. MR**84b:22029****[K]**A. Kleshchev, Branching rules for modular representations of symmetric groups II,*J. Reine Angew. Math.***459**(1995), 163-212. MR**96m:20019b****[K]**A. Kleshchev, Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineux,*J. London Math. Soc.***54**(1996), 25-38. MR**96m:20019c****[K]**A. Kleshchev, Branching rules for modular representations of symmetric groups IV,*J. Algebra***201**(1998), 547-572. MR**99d:20019****[K]**A. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups,*Proc. London Math. Soc.***75**(1997), 497-558. MR**98g:20026****[LLT]**A. Lascoux, B. Leclerc and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras,*Comm. Math. Phys.***181**(1996), 205-263. MR**97k:17019****[LT]**B. Leclerc and J.-Y. Thibon, Canonical bases of -deformed Fock spaces,*Internat. Math. Res. Notices***9**(1996), 447-456. MR**97h:17023****[LT]**B. Leclerc and J.-Y. Thibon, -Deformed Fock spaces and modular representations of spin symmetric groups,*J. Phys. A***30**(1997), 6163-6176. MR**99c:17029****[Le]**D.A. Leites, Introduction to the theory of supermanifolds,*Russian Math. Surveys***35**(1980), 1-64.**[ML]**S. MacLane,*Categories for the working mathematician*, Graduate Texts in Mathematics 5, Springer-Verlag, 1971. MR**50:7275****[Man]**Yu I. Manin,*Gauge field theory and complex geometry*, Grundlehren der Mathematischen Wissenschaften 289, second edition, Springer, 1997. MR**99e:32001****[MM]**K. Misra and T. Miwa, Crystal base for the basic representation of ,*Comm. Math. Phys.***134**(1990), 79-88. MR**91j:17021****[Mo]**A.O. Morris, The spin representations of the symmetric group,*Canad. J. Math.***17**(1965), 543-549. MR**31:240**involving shifted Young diagrams,*Math. Proc. Camb. Phil. Soc.***99**(1986), 23-31.**[N]**M. Nazarov, Young's symmetrizers for projective representations of the symmetric group,*Advances Math.***127**(1997), 190-257. MR**98m:20019****[OV]**A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups.*Selecta Math. (N.S.)***2**(1996), 581-605. MR**99g:20024****[O]**G. Olshanski, Quantized universal enveloping superalgebra of type and a super-extension of the Hecke algebra,*Lett. Math. Phys.***24**(1992), 93-102. MR**93i:17004****[Sch]**O. Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces,*Internat. Math. Res. Notices***8**(2000), 413-440. MR**2001g:16032****[S]**A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras and ,*Math. USSR Sbornik***51**(1985), 419-427. MR**2000j:20021****[S]**A. N. Sergeev, The Howe duality and the projective representations of symmetric groups,*Represent. Theory***3**(1999), 416-434. MR**2001c:17029****[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. Zelevinsky, Induced representations of reductive -adic groups, II,*Ann. Sci. Ecole Norm. Sup.***13**(1980), 165-210. MR**83g:22012****[Z]**A. Zelevinsky,*Representations of finite classical groups*, Lecture Notes in Math. 869, Springer-Verlag, Berlin, 1981. MR**83k:20017**

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
17B67,
20C08,
20C20,
17B10,
17B37

Retrieve articles in all journals with MSC (2000): 17B67, 20C08, 20C20, 17B10, 17B37

Additional Information

**Jonathan Brundan**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Email:
brundan@darkwing.uoregon.edu

**Alexander Kleshchev**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Email:
klesh@math.uoregon.edu

DOI:
https://doi.org/10.1090/S1088-4165-01-00123-6

Received by editor(s):
March 9, 2001

Received by editor(s) in revised form:
August 15, 2001

Published electronically:
October 24, 2001

Additional Notes:
Both authors were partially supported by the NSF (grant nos DMS-9801442 and DMS-9900134)

Article copyright:
© Copyright 2001
American Mathematical Society