Crystal structures arising from representations of $GL(m|n)$
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- by Jonathan Kujawa
- Represent. Theory 10 (2006), 49-85
- DOI: https://doi.org/10.1090/S1088-4165-06-00219-6
- Published electronically: February 16, 2006
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Abstract:
This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that $GL(m|n)$ has non-conjugate Borel subgroups and we show how Serganova’s odd reflections give rise to canonical crystal isomorphisms.References
- Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m|n)$, J. Amer. Math. Soc. 16 (2003), no. 1, 185–231. MR 1937204, DOI 10.1090/S0894-0347-02-00408-3
- 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, On translation functors for general linear and symmetric groups, Proc. London Math. Soc. (3) 80 (2000), no. 1, 75–106. MR 1719176, DOI 10.1112/S0024611500012132
- Jonathan Brundan and Jonathan Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), no. 1, 13–39. MR 2002217, DOI 10.1023/A:1025113308552
- Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. MR 354887, DOI 10.1007/BF01214125
- V. K. Dobrev and V. B. Petkova, Group-theoretical approach to extended conformal supersymmetry: function space realization and invariant differential operators, Fortschr. Phys. 35 (1987), no. 7, 537–572. MR 905398, DOI 10.1002/prop.2190350705
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
- 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
- A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI 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 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 10.1006/jabr.1995.1362 thesis J. Kujawa, The representation theory of the supergroup $GL(m|n)$, Ph.D. thesis, University of Oregon, 2003. Leites D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1–64.
- Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$, Selecta Math. (N.S.) 2 (1996), no. 4, 607–651. MR 1443186, DOI 10.1007/PL00001385
- Vera Serganova, Kazhdan-Lusztig polynomials for Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 151–165. MR 1237837 sergthesis V. Serganova, Automorphisms of complex simple Lie superalgebras and affine Kac-Moody algebras, Ph.D. thesis, Leningrad State University, 1988. sergeev2 A. Sergeev, Enveloping algebra centre for Lie superalgebras GL and Q, Ph.D. thesis, Moscow State University, Moscow, 1987.
- Alexander Sergeev, The invariant polynomials on simple Lie superalgebras, Represent. Theory 3 (1999), 250–280. MR 1714627, DOI 10.1090/S1088-4165-99-00077-1
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
Bibliographic Information
- Jonathan Kujawa
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 720815
- Email: kujawa@math.uga.edu
- Received by editor(s): November 17, 2003
- Received by editor(s) in revised form: January 3, 2006
- Published electronically: February 16, 2006
- Additional Notes: Research was supported in part by NSF grant DMS-0402916
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 10 (2006), 49-85
- MSC (2000): Primary 20C20, 05E99; Secondary 17B10
- DOI: https://doi.org/10.1090/S1088-4165-06-00219-6
- MathSciNet review: 2209849