Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Generalized exponents of small representations. I


Author: Bogdan Ion
Journal: Represent. Theory 13 (2009), 401-426
MSC (2000): Primary 17B10
DOI: https://doi.org/10.1090/S1088-4165-09-00359-8
Published electronically: September 10, 2009
MathSciNet review: 2540703
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is the first paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. The main part of this paper illustrates the overall structure of the argument on root systems of type $ A$ and discusses the relationship with the Lascoux-Schützenberger charge formula


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

  • 1. W. M. BEYNON and G. LUSZTIG, Some numerical results on the characters of exceptional Weyl groups. Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 417-426. MR 503002 (80a:20017)
  • 2. A. BROER, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants. Indag. Math. (N.S.) 6 (1995), no. 4, 385-396. MR 1365182 (96j:20058)
  • 3. R. K. BRYLINSKI, Limits of weight spaces, Lusztig's $ q$-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc. 2 (1989), no. 3, 517-533. MR 984511 (90g:17011)
  • 4. L. M. BUTLER, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc. 112 (1994), no. 539. MR 1223236 (95e:05122)
  • 5. I. M. GESSEL. Multipartite $ P$-partitions and inner products of skew Schur functions. Combinatorics and algebra (Boulder, Colo., 1983), 289-317, Contemp. Math. 34, Amer. Math. Soc., Providence, RI, 1984. MR 777705 (86k:05007)
  • 6. E. A. GUTKIN, Representations of the Weyl group in the space of vectors of zero weight. Uspehi Mat. Nauk 28 (1973), no. 5 (173), 237-238. MR 0409671 (53:13423)
  • 7. B. ION, The Cherednik kernel and generalized exponents. Int. Math. Res. Not. 2004, no. 36, 1869-1895. MR 2058356 (2005a:17004)
  • 8. B. ION, Generalized exponents of small representations. II. http://arxiv.org/abs/0904.2487
  • 9. B. ION, Generalized exponents of small representations. III. In preparation.
  • 10. B. KOSTANT, Lie group representations on polynomial rings. Amer. J. Math. 85 (1963), 327-404. MR 0158024 (28:1252)
  • 11. B. KOSTANT, On Macdonald's $ \eta $-function formula, the Laplacian and generalized exponents. Adv. Math. 20 (1976), no. 2, 179-212. MR 0485661 (58:5484)
  • 12. A. LASCOUX AND M.-P. SCHÜTZENBERGER, Sur une conjecture de H. O. Foulkes. C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323-A324. MR 0472993 (57:12672)
  • 13. G. LUSZTIG, Irreducible representations of finite classical groups. Invent. Math. 43 (1977), no. 2, 125-175. MR 0463275 (57:3228)
  • 14. G. LUSZTIG, Singularities, character formulas, and a $ q$-analog of weight multiplicities. Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque 101-102, 208-229, Soc. Math. France, Paris, 1983. MR 737932 (85m:17005)
  • 15. I. G. MACDONALD, Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • 16. M. REEDER, Zero weight spaces and the Springer correspondence. Indag. Math. (N.S.) 9 (1998), no. 3, 431-441. MR 1692153 (2000e:22005)
  • 17. R. STEINBERG, A geometric approach to the representations of the full linear group over a Galois field. Trans. Amer. Math. Soc. 71 (1951), 274-282. MR 0043784 (13:317d)
  • 18. J. R. STEMBRIDGE, First layer formulas for characters of $ {\rm SL}(n, \C)$. Trans. Amer. Math. Soc. 299 (1987), no. 1, 319-350. MR 869415 (88g:20088)
  • 19. J. R. STEMBRIDGE, On the eigenvalues of representations of reflection groups and wreath products. Pacific J. Math. 140 (1989), no. 2, 353-396. MR 1023791 (91a:20022)
  • 20. J. R. STEMBRIDGE, Graded multiplicities in the Macdonald kernel. I. IMRP Int. Math. Res. Pap. 2005 no. 4, 183-236. MR 2199453 (2006m:05257)
  • 21. D. A. VOGAN, Associated varieties and unipotent representations. Harmonic analysis on reductive groups (Brunswick, ME, 1989), 315-388, Progr. Math. 101, Birkhäuser Boston, Boston, MA, 1991. MR 1168491 (93k:22012)

Similar Articles

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

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


Additional Information

Bogdan Ion
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260; Algebra and Number Theory Research Center, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei St., Bucharest, Romania
Email: bion@pitt.edu

DOI: https://doi.org/10.1090/S1088-4165-09-00359-8
Received by editor(s): April 18, 2009
Received by editor(s) in revised form: June 26, 2009
Published electronically: September 10, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society