## 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, - 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, - 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**

*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.

*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.

## 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