Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs


Author: Francesco Brenti
Journal: Trans. Amer. Math. Soc. 361 (2009), 1703-1729
MSC (2000): Primary 05E99; Secondary 20F55
DOI: https://doi.org/10.1090/S0002-9947-08-04458-9
Published electronically: October 29, 2008
MathSciNet review: 2465813
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. In particular, we show that these polynomials are always either zero or a monic power of $ q$, and that they are combinatorial invariants.


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

  • 1. A. Björner, private communication, March 1992.
  • 2. A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer-Verlag, New York, 2005. MR 2133266 (2006d:05001)
  • 3. B. Boe, Kazhdan-Lusztig polynomials for Hermitian symmetric spaces, Trans. Amer. Math. Soc., 309 (1988), 279-294. MR 957071 (89i:22024)
  • 4. F. Brenti, Kazhdan-Lusztig and $ R$-polynomials, Young's lattice, and Dyck partitions, Pacific J. Math., 207 (2002), 257-286. MR 1972246 (2004e:20008)
  • 5. F. Brenti, Parabolic Kazhdan-Lusztig $ R$-polynomials for Hermitian symmetric pairs, J. Algebra, 318 (2007), 412-429. MR 2363142
  • 6. L. Casian, D. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Zeit., 195 (1987), 581-600. MR 900346 (88i:17008)
  • 7. V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra, 111 (1987), 483-506. MR 916182 (89a:20054)
  • 8. V. Deodhar, Duality in parabolic setup for questions in Kazhdan-Lusztig theory, J. Algebra, 142 (1991), 201-209. MR 1125213 (92j:20049)
  • 9. J. Haglund, M. Haiman, N. Loehr, A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc., 18 (2005), 735-761. MR 2138143 (2006g:05223a)
  • 10. J. Haglund, M. Haiman, N. Loehr, J. Remmel, A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., 126 (2005), 195-232. MR 2115257 (2006f:05186)
  • 11. J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, no. 29, Cambridge Univ. Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • 12. M. Kashiwara, T. Tanisaki, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, J. Algebra, 249 (2002), 306-325. MR 1901161 (2004a:14049)
  • 13. D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184. MR 560412 (81j:20066)
  • 14. B. Leclerc, J.-Y. Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, Adv. Studies Pure Math., 28 (2000), 155-220. MR 1864481 (2002k:20014)
  • 15. W. Soergel, Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Represent. Theory, 1 (1997), 83-114. MR 1444322 (98d:17026)
  • 16. W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory, 1 (1997), 115-132. MR 1445716 (98f:17016)
  • 17. R. P. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
  • 18. R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics, no. 62, Cambridge Univ. Press, Cambridge, 1999. MR 1676282 (2000k:05026)
  • 19. M. Varagnolo, E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J., 100 (1999), 267-297. MR 1722955 (2001c:17029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E99, 20F55

Retrieve articles in all journals with MSC (2000): 05E99, 20F55


Additional Information

Francesco Brenti
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy
Email: brenti@mat.uniroma2.it

DOI: https://doi.org/10.1090/S0002-9947-08-04458-9
Received by editor(s): August 7, 2006
Received by editor(s) in revised form: November 9, 2006
Published electronically: October 29, 2008
Additional Notes: The author was partially supported by EU grant HPRN-CT-2001-00272. Part of this research was carried out while the author was a member of the Mittag-Leffler Institut in Djürsholm, Sweden, whose hospitality and financial support are gratefully acknowledged.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society