Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T05:44:07.491Z Has data issue: false hasContentIssue false
  • English
  • Français

A GRAPHICAL DESCRIPTION OF (Dn,An−1) KAZHDAN–LUSZTIG POLYNOMIALS

Published online by Cambridge University Press:  02 August 2012

TOBIAS LEJCZYK  and
CATHARINA STROPPEL
TOBIAS LEJCZYK
Affiliation:
Department of Mathematics, Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected]
CATHARINA STROPPEL
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.

Keywords

17B1016G1005E15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 313 - 340
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Beilinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, JAMS 2 (9) (1996), 473–527.Google Scholar
2.Billey, S. C. and Jones, B. C., Embedded factor patterns for Deodhar elements in Kazhdan–Lusztig theory, Ann. Comb. 11 (3) (2007), 95–119.Google Scholar
3.Billey, S. C. and Warrington, G. S., Kazhdan–Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebr. Comb. 13 (2) (2001), 111–136.CrossRefGoogle Scholar
4.Björner, A. and Brenti, F., Combinatorics of coxeter groups, Graduate Texts in Math, vol. 231 (Springer, New York, 2005).Google Scholar
5.Boe, B. D., Kazhdan–Lusztig polynomials for Hermitian symmetric spaces, Trans. AMS 309 (1) (1988), 279–294.CrossRefGoogle Scholar
6.Braden, T., Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002), 493–532.Google Scholar
7.Brenti, F., Kazhdan–Lusztig polynomials for Hermition symmetric pairs, Trans. AMS 361 (4) (2009), 1703–1729.Google Scholar
8.Brundan, J. and Stroppel, C., Gradings on walled Brauer algebras and Khovanov's arc algebra, Adv. Math. (to appear) arXiv:1107.0999 (2011).CrossRefGoogle Scholar
9.Brundan, J. and Stroppel, C., Highest weight categories arising from Khovanov's diagram algebra I: Cellularity, Mosc. Math. J. 11, (2011) 685–722.CrossRefGoogle Scholar
10.Brundan, J. and Stroppel, C., Highest weight categories arising from Khovanov's diagram algebra III: Category, Repr. Theory 15 (2011), 170–243.Google Scholar
11.Brundan, J. and Stroppel, C., Highest weight categories arising from Khovanov's diagram algebra IV: The general linear supergroup, JEMS 14 (2012), 373–419.Google Scholar
12.Casian, L. G. and Collingwood, D. H., The Kazhdan–Lusztig conjecture for generalized Verma modules, Math. Zeitschrift 195 (4) (1987), 581–600.CrossRefGoogle Scholar
13.Cheng, S.-J., Lam, N. and Wang, W., Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (1) (2011), 189–224.Google Scholar
14.Cox, A. and De, M.Visscher, Diagrammatic Kazhdan–Lusztig theory for the (walled) Brauer algebra, J. Algebra 340 (2011), 151–181.Google Scholar
15.Deodhar, V. V., A combinatorial setting for questions in Kazhdan–Lusztig theory, Geom. Dedicata 36 (1) (1990), 95–119.Google Scholar
16.Ehrig, M. and Stroppel, C., Diagram calculus for perverse sheaves on the isotropic Grassmannians and Springer fibers of type D (in preparation).Google Scholar
17.Ehrig, M. and Stroppel, C., A graded version of the Brauer algebra (in the non-semisimple case) (in preparation).Google Scholar
18.Ehrig, M. and Stroppel, C., Springer fibers of type D from a topological point of view (in preparation ).Google Scholar
19.Graham, J. J. and Lehrer, G. I., Cellular algebras, Invent. Math. 123 (1996), 1–34.Google Scholar
20.Green, R. M., Generalized Temperley-Lieb algebras and decorated tangles, JKTR 7 (2) (1998), 155–177.Google Scholar
21.Gruson, C. and Serganova, V., Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. London Math. Soc. 101 (3) (2010), 852–892.Google Scholar
22.Humphreys, J. E., Reflection groups and coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, UK, 1992).Google Scholar
23.Humphreys, J. E., Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, vol. 94 (AMS, Providence, RI, 2008).Google Scholar
24.Kazhdan, D. and Lusztig, G., Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (2) (1979), 165–184.CrossRefGoogle Scholar
25.Khovanov, M., A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359–426.Google Scholar
26.Lejczyk, T., A graphical description of (A n−1, Dn) Kazhdan–Lusztig polynomials, Diploma Thesis (University of Bonn, Bonn, Germany, 2010).Google Scholar
27.Martin, P., The decomposition matrices of the Brauer algebra over the complex field, arXiv:0908.1500 (2009).Google Scholar
28.Mathas, A., Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15 (AMS, Providence, RI, 1999).Google Scholar
29.Schäfer, G., A graphical calculus for 2-block Spaltenstein varieties, Glasg. Math. J. 54 (2012), 449–477.Google Scholar
30.Serre, J. P., Linear representations of finite groups, vol. 42 (Springer, New York, 1977).CrossRefGoogle Scholar
31.Shigechi, K. and Zinn-Justin, P., Path representation of maximal parabolic Kazhdan-Lusztig polynomials, J. Pure Appl. Alg. 216 (11) (2012), 2533–2548.Google Scholar
32.Soergel, W., Kazhdan–Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Rep. Theory 1 (1997), 37–68.Google Scholar
33.Stroppel, C., Category : Gradings and translation functors, J. Algebra 268 (1) (2003), 301–326.CrossRefGoogle Scholar
34.Stroppel, C., Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (3) (2005), 547–596.Google Scholar
35.Stroppel, C., Parabolic category , perverse sheaves on Grassmanninans, Springer fibres and Khovanov homology, Comp. Math. 145 (2009), 954–992.Google Scholar