Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak{gl}(m\vert n)$


Author: Jonathan Brundan
Journal: J. Amer. Math. Soc. 16 (2003), 185-231
MSC (2000): Primary 17B10
DOI: https://doi.org/10.1090/S0894-0347-02-00408-3
Published electronically: October 16, 2002
MathSciNet review: 1937204
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m\vert n)$, and determine ${\operatorname{Ext}}$'s between simple modules in the category of finite dimensional representations. We formulate conjectures for the analogous results in category $\mathcal O$. The combinatorics parallels the combinatorics of certain canonical bases over the Lie algebra $\mathfrak{gl}(\infty)$.


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

  • [BB] A. Beilinson and J. Bernstein, Localisation de $\mathfrak g$-modules, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15-18. MR 82k:14015
  • [BGS] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. MR 96k:17010
  • [BR] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Advances Math. 64 (1987), 118-175. MR 88l:20006
  • [BGG] J. Bernstein, I. M. Gelfand and S. I. Gelfand, A category of $\mathfrak g$-modules, Func. Anal. Appl. 10 (1976), 87-92. MR 53:10880
  • [BL] J. Bernstein and D. Leites, Character formulae for irreducible representations of Lie superalgebras of series $\mathfrak{gl}$ and $\mathfrak{sl}$, C. R. Acad. Bulg. Sci. 33 (1980), 1049-1051. MR 82j:17020a
  • [Bou] N. Bourbaki, Commutative algebra, Springer, 1989. MR 90a:13001
  • [B1] J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type $\mathbf A$, Proc. London Math. Soc. 77 (1998), 551-581. MR 2000d:20007
  • [B2] J. Brundan, Tilting modules for Lie superalgebras, preprint, University of Oregon, 2002, available from http://darkwing.uoregon.edu/$\sim$brundan/research.html.
  • [BK] J. Brundan and A. Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2\ell}^{(2)}$ and modular branching rules for $\widehat{S}_n$, Represent. Theory 5 (2001), 317-403. MR 2002j:17024
  • [BKu] J. Brundan and J. Kujawa, A new proof of the Mullineux conjecture, preprint, University of Oregon, 2002.
  • [BrK] J. L. Brylinksi and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 83e:22020
  • [CPS1] E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 85-99. MR 90d:18005
  • [CPS2] E. Cline, B. Parshall and L. Scott, Duality in highest weight categories, Contemp. Math. 82 (1989), 7-22. MR 90g:17014
  • [CPS3] E. Cline, B. Parshall and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. J. 45 (1993), 511-534. MR 94k:20079
  • [CPS4] E. Cline, B. Parshall and L. Scott, Infinitesimal Kazhdan-Lusztig theories, Contemp. Math. 139 (1992), 43-73. MR 93j:20089
  • [Deo] V. Deodhar, On some geometric aspects of Bruhat orderings II: the parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), 483-506. MR 89a:20054
  • [Dix] J. Dixmier, Enveloping algebras, Graduate Studies in Math. 11, Amer. Math. Soc., 1996. MR 97c:17010
  • [D1] J. Du, IC bases and quantum linear groups, Proc. Symp. Pure Math. 56 (1994), Part 2, 135-148. MR 95d:17010
  • [D2] J. Du, A note on quantized Weyl reciprocity at roots of unity, Alg. Colloq. 2 (1995), 363-372. MR 96m:17024
  • [FKK] I. B. Frenkel, M. Khovanov and A. A. Kirillov, Jr., Kazhdan-Lusztig polynomials and canonical basis, Transform. Groups 3 (1998), 321-336. MR 2000f:20071
  • [HKJ] J. W. B. Hughes, R. C. King and J. van der Jeugt, On the composition factors of Kac modules for the Lie superalgebra $\mathfrak{sl}(m\vert n)$, J. Math. Phys. 33 (1992), 470-491. MR 93a:17003
  • [J1] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math. no. 750, Springer-Verlag, 1983. MR 81m:17011
  • [J2] J. C. Jantzen, Representations of algebraic groups, Academic Press, 1987. MR 89c:20001
  • [JHKT1] J. van der Jeugt, J. W. B. Hughes, R. C. King and J. Thierry-Mieg, Character formulas for irreducible modules of the Lie superalgebra $\mathfrak{sl}(m\vert n)$, J. Math. Phys. 31 (1990), 2278-2304. MR 92b:17049
  • [JHKT2] J. van der Jeugt, J. W. B. Hughes, R. C. King and J. Thierry-Mieg, A character formula for singly atypical modules of the Lie superalgebra $\mathfrak{sl}(m\vert n)$, Commun. Alg. 18 (1990), 3453-3480. MR 91j:17046
  • [JZ] J. van der Jeugt and R. B. Zhang, Characters and composition factor multiplicities for the Lie superalgebra $\mathfrak{gl}(m\vert n)$, Lett. Math. Phys. 47 (1999), 49-61. MR 2000a:17008
  • [Ka1] V. Kac, Lie superalgebras, Advances in Math. 26 (1977), 8-96. MR 58:5803
  • [Ka2] V. Kac, Characters of typical representations of classical Lie superalgebras, Commun. in Algebra 5 (8) (1977), 889-897. MR 56:3075
  • [Ka3] V. Kac, Representations of classical Lie superalgebras, in: ``Differential geometrical methods in mathematical physics II'', Lecture Notes in Math. no. 676, pp. 597-626, Springer-Verlag, Berlin, 1978. MR 80f:17006
  • [KaW] V. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Progress Math. 123 (1994), 415-456. MR 96j:11056
  • [K1] M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455-485. MR 94b:17024
  • [K2] M. Kashiwara, On crystal bases, Proc. Canadian Math. Soc. 16 (1995), 155-196. MR 97a:17016
  • [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • [Kv] A. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. reine angew. Math. 459 (1995), 163-212. MR 96m:20019
  • [Ku] J. Kujawa, The representation theory of the supergroup $GL(m\vert n)$, PhD thesis, University of Oregon, 2003.
  • [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
  • [L] G. Lusztig, Introduction to quantum groups, Progress in Math. 110, Birkhäuser, 1993. MR 94m:17016
  • [Mac] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition, OUP, 1995. MR 96h:05207
  • [M] Yu I. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften 289, second edition, Springer, 1997. MR 99e:32001
  • [PS1] I. Penkov and V. Serganova, Representations of classical Lie superalgebras of type I, Indag. Math. (N. S.) 3 (1992), 419-466. MR 93k:17006
  • [PS2] I. Penkov and V. Serganova, Generic irreducible representations of finite dimensional Lie superalgebras, Internat. J. Math. 5 (1994), 389-419. MR 95c:17015
  • [S1] V. Serganova, Automorphisms of complex simple Lie superalgebras and affine Kac-Moody algebras, PhD thesis, Leningrad State University, 1988.
  • [S2] V. Serganova, Kazhdan-Lusztig polynomials for the Lie superalgebra $GL(m\vert n)$, Adv. Sov. Math. 16 (1993), 151-165. MR 94k:17005
  • [S3] V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra $\mathfrak{gl}(m\vert n)$, Selecta Math. 2 (1996), 607-651. MR 98f:17007
  • [Sg1] A. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras $GL(n,m)$ and $Q(n)$, Math. USSR Sbornik 51 (1985), 419-427. MR 85h:17010
  • [Sg2] A. Sergeev, Enveloping algebra centre for Lie superalgebras $GL$ and $Q$, PhD thesis, Moscow State University, 1987.
  • [Sg3] A. Sergeev, The invariant polynomials on simple Lie superalgebras, Represent. Theory 3 (1999), 250-280. MR 2000k:17012
  • [So1] W. Soergel, Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Represent. Theory 1 (1997), 83-114. MR 98d:17026
  • [So2] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-448. MR 2000c:17048
  • [V] D. Vogan, Irreducible representations of semisimple Lie groups II: the Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), 805-859. MR 81f:22024
  • [W] C. Weibel, An introduction to homological algebra, CUP, 1994. MR 95f:18001
  • [Z] Y. M. Zou, Categories of finite dimensional weight modules over type I classical Lie superalgebras, J. Algebra 180 (1996), 459-482. MR 97e:17012

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 17B10

Retrieve articles in all journals with MSC (2000): 17B10


Additional Information

Jonathan Brundan
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: brundan@darkwing.uoregon.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00408-3
Received by editor(s): March 12, 2002
Received by editor(s) in revised form: September 25, 2002
Published electronically: October 16, 2002
Additional Notes: Research partially supported by the NSF (grant no. DMS-0139019)
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society