Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity
Ewan Delanoy
Université Lyon 1 Institut Camille Jordan, UMR 5208 CNRS 69622 Villeurbanne Cedex France 69622 Villeurbanne Cedex France
DOI: 10.1007/s10801-006-0014-7
Abstract
We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [ e, u] only depend on the isomorphism type of [ e, u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for special matchings discovered by Brenti and the general special matching machinery developed by Du Cloux.
Pages: 437–463
Keywords: keywords Coxeter group; Kazhdan-Lusztig polynomials; special matching
Full Text: PDF
References
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2. F. Brenti, “The intersection cohomology of Schubert varieties is a combinatorial invariant,” Europ. J. Combin. 25 (2004), 1151-1167.
3. F. Brenti, F.Caselli, and M. Marietti, “Special matchings and Kazhdan-Lusztig polynomials,” Adv. in Math. To appear, available on http://www.mat.uniroma2.it/ brenti/papers.htm
4. F. du Cloux, “An abstract model for Bruhat intervals,” Europ. J. Combin. 21 (2000), 197-222.
5. F. du Cloux, Coxeter, version beta. Available on http://www.desargues.univlyon1.fr/home/ducloux/coxeter.html
6. F. du Cloux, “Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials,” Adv. in Math. 180 (2003), 146-175.
7. F. du Cloux, “A transducer approach to Coxeter groups,” J. Symb. Comp, 27 (1999), 1-14.
8. M. Dyer, Hecke Algebras and reflections in Coxeter groups, PhD thesis, University of Sydney, 1987.
9. J.E. Humphreys, Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge, 1990.
10. D. Kazhdan and G. Lusztig, “Representations of Coxeter Groups and Hecke Algebras,” Invent. Math. 53 (1979), 165-184.
11. M. Marietti, “Kazhdan-Lusztig Theory: Boolean elements, special matchings and combinatorial invariance,” Ph.D. Thesis, University of Rome ”La Sapienza”, 2003.
12. W.C. Waterhouse, “Automorphisms of the Bruhat ordering on Coxeter groups,” Bull. London Math. Soc., 21 (1989), 243-248.