Special matchings and parabolic Kazhdan–Lusztig polynomials
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- by Mario Marietti PDF
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Abstract:
We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan–Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in Advances in Math. 202 (2006), 555-601. As a consequence, the parabolic Kazhdan–Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal coset representatives.References
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- Francesco Brenti, Fabrizio Caselli, and Mario Marietti, Special matchings and Kazhdan-Lusztig polynomials, Adv. Math. 202 (2006), no. 2, 555–601. MR 2222360, DOI 10.1016/j.aim.2005.01.011
- Fabrizio Caselli and Mario Marietti, Special matchings in Coxeter groups, preprint.
- Vinay V. Deodhar, Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), no. 2, 187–198. MR 435249, DOI 10.1007/BF01390109
- Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182, DOI 10.1016/0021-8693(87)90232-8
- Matthew Dyer, Hecke algebras and reflections in Coxeter groups, Ph. D. Thesis, University of Sydney, 1987.
- Matthew Dyer, On the “Bruhat graph” of a Coxeter system, Compositio Math. 78 (1991), no. 2, 185–191. MR 1104786
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089–1136. MR 3245013, DOI 10.4007/annals.2014.180.3.6
- A. van den Hombergh, About the automorphisms of the Bruhat-ordering in a Coxeter group, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 125–131. MR 0360857
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
- P. Mongelli, Coxeter groups: statistics and Kazhdan–Lusztig polynomials, Ph. D. Thesis, Sapienza - University of Rome, 2012.
- Richard P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo Rota. MR 847717, DOI 10.1007/978-1-4615-9763-6
- Jacques Tits, Le problème des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome, 1967/68) Academic Press, London, 1969, pp. 175–185 (French). MR 0254129
Additional Information
- Mario Marietti
- Affiliation: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
- MR Author ID: 689090
- Email: m.marietti@univpm.it
- Received by editor(s): May 8, 2014
- Received by editor(s) in revised form: July 20, 2014, September 14, 2014, and November 23, 2014
- Published electronically: June 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 5247-5269
- MSC (2010): Primary 05E99, 20F55
- DOI: https://doi.org/10.1090/tran6676
- MathSciNet review: 3456179