Conjectures About Certain Parabolic Kazhdan–Lusztig Polynomials

  • Erez LapidEmail author
Conference paper
Part of the Simons Symposia book series (SISY)


Irreducibility results for parabolic induction of representations of the general linear group over a local non-Archimedean field can be formulated in terms of Kazhdan–Lusztig polynomials of type A. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan–Lusztig polynomials known as parabolic Kazhdan–Lusztig polynomials satisfy properties analogous to those of the ordinary ones.


Kazhdan–Lusztig polynomials 



The author would like to thank Karim Adiprasito, Joseph Bernstein, Sara Billey, David Kazhdan, George Lusztig, Greg Warrington, Geordie Williamson, and Zhiwei Yun for helpful correspondence. We also thank the referee for useful suggestions.


  1. [BB05]
    Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266 (2006d:05001)Google Scholar
  2. [BBL98]
    Prosenjit Bose, Jonathan F. Buss, and Anna Lubiw, Pattern matching for permutations, Inform. Process. Lett. 65 (1998), no. 5, 277–283. MR 1620935MathSciNetCrossRefGoogle Scholar
  3. [BC17]
    Francesco Brenti and Fabrizio Caselli, Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials, Adv. Math. 304 (2017), 539–582. MR 3558217MathSciNetCrossRefGoogle Scholar
  4. [BH99]
    Brigitte Brink and Robert B. Howlett, Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999), no. 2, 323–351. MR 1688445MathSciNetCrossRefGoogle Scholar
  5. [BJS93]
    Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505Google Scholar
  6. [BMB07]
    Mireille Bousquet-Mélou and Steve Butler, Forest-like permutations, Ann. Comb. 11 (2007), no. 3-4, 335–354. MR 2376109MathSciNetCrossRefGoogle Scholar
  7. [BMS16]
    Francesco Brenti, Pietro Mongelli, and Paolo Sentinelli, Parabolic Kazhdan-Lusztig polynomials for quasi-minuscule quotients, Adv. in Appl. Math. 78 (2016), 27–55. MR 3497995Google Scholar
  8. [Bor98]
    Richard E. Borcherds, Coxeter groups, Lorentzian lattices, and K3surfaces, Internat. Math. Res. Notices (1998), no. 19, 1011–1031. MR MR1654763 (2000a:20088)Google Scholar
  9. [Bre02]
    Francesco Brenti, Kazhdan-Lusztig and R-polynomials, Young’s lattice, and Dyck partitions, Pacific J. Math. 207 (2002), no. 2, 257–286. MR 1972246MathSciNetCrossRefGoogle Scholar
  10. [BW01]
    Sara C. Billey and Gregory S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin. 13 (2001), no. 2, 111–136. MR 1826948Google Scholar
  11. [BW03]
    ——, Maximal singular loci of Schubert varieties in SL(n)∕B, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3915–3945. MR 1990570Google Scholar
  12. [BY13]
    Roman Bezrukavnikov and Zhiwei Yun, On Koszul duality for Kac-Moody groups, Represent. Theory 17 (2013), 1–98. MR 3003920MathSciNetCrossRefGoogle Scholar
  13. [dC02]
    Fokko du Cloux, Computing Kazhdan-Lusztig polynomials for arbitrary Coxeter groups, Experiment. Math. 11 (2002), no. 3, 371–381. MR 1959749 (2004j:20084)Google Scholar
  14. [Dem74]
    Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I. MR 0354697 (50 #7174)Google Scholar
  15. [Deo85]
    Vinay V. Deodhar, Local Poincaré duality and nonsingularity of Schubert varieties, Comm. Algebra 13 (1985), no. 6, 1379–1388. MR 788771 (86i:14015)MathSciNetCrossRefGoogle Scholar
  16. [Deo87]
    ——, 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 (89a:20054)Google Scholar
  17. [Deo90]
    ——, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata 36 (1990), no. 1, 95–119. MR 1065215 (91h:20075)Google Scholar
  18. [Deo94]
    Vinay Deodhar, A brief survey of Kazhdan-Lusztig theory and related topics, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 105–124. MR 1278702 (96d:20039)Google Scholar
  19. [EW14]
    Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089–1136. MR 3245013Google Scholar
  20. [Fan98]
    C. K. Fan, Schubert varieties and short braidedness, Transform. Groups 3 (1998), no. 1, 51–56. MR 1603806 (98m:14052)MathSciNetCrossRefGoogle Scholar
  21. [FG97]
    C. K. Fan and R. M. Green, Monomials and Temperley-Lieb algebras, J. Algebra 190 (1997), no. 2, 498–517. MR 1441960 (98a:20037)MathSciNetCrossRefGoogle Scholar
  22. [Fox]
    Jacob Fox, Stanley-Wilf limits are typically exponential, Adv. Math. to appear, arXiv:1310.8378.Google Scholar
  23. [GM14]
    Sylvain Guillemot and Dániel Marx, Finding small patterns in permutations in linear time, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2014, pp. 82–101. MR 3376367Google Scholar
  24. [Hen07]
    Anthony Henderson, Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A, Represent. Theory 11 (2007), 95–121 (electronic). MR 2320806MathSciNetCrossRefGoogle Scholar
  25. [How80]
    Robert B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. (2) 21 (1980), no. 1, 62–80. MR 576184Google Scholar
  26. [JW13]
    Brant Jones and Alexander Woo, Mask formulas for cograssmannian Kazhdan-Lusztig polynomials, Ann. Comb. 17 (2013), no. 1, 151–203. MR 3027577MathSciNetCrossRefGoogle Scholar
  27. [KL79]
    David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412 (81j:20066)MathSciNetCrossRefGoogle Scholar
  28. [KT02]
    Masaki Kashiwara and Toshiyuki Tanisaki, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, J. Algebra 249 (2002), no. 2, 306–325. MR 1901161 (2004a:14049)MathSciNetCrossRefGoogle Scholar
  29. [Lap17]
    Erez Lapid, A tightness property of relatively smooth permutations, 2017, arXiv:1710.06115.Google Scholar
  30. [Las95]
    Alain Lascoux, Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 667–670. MR 1354702 (96g:05144)Google Scholar
  31. [LM16]
    Erez Lapid and Alberto Mínguez, Geometric conditions for \(\square \) -irreducibility of certain representations of the general linear group over a non-Archimedean local field, 2016, arXiv:1605.08545.Google Scholar
  32. [LS90]
    V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in Sl(n)∕B, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 1, 45–52. MR 1051089MathSciNetCrossRefGoogle Scholar
  33. [Lus93]
    G. Lusztig, Tight monomials in quantized enveloping algebras, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 117–132. MR 1261904Google Scholar
  34. [Lus03]
    ——, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442Google Scholar
  35. [Lus77]
    ——, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101–159. MR 0453885Google Scholar
  36. [LW17]
    Nicolas Libedinsky and Geordie Williamson, The anti-spherical category, 2017, arXiv:1702.00459.Google Scholar
  37. [Mac04]
    Percy A. MacMahon, Combinatory analysis. Vol. I, II (bound in one volume), Dover Phoenix Editions, Dover Publications, Inc., Mineola, NY, 2004, Reprint of An introduction to combinatory analysis (1920) and Combinatory analysis. Vol. I, II (1915, 1916). MR 2417935Google Scholar
  38. [Mon14]
    Pietro Mongelli, Kazhdan-Lusztig polynomials of Boolean elements, J. Algebraic Combin. 39 (2014), no. 2, 497–525. MR 3159260Google Scholar
  39. [MT04]
    Adam Marcus and Gábor Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2004), no. 1, 153–160. MR 2063960Google Scholar
  40. [Sen14]
    Paolo Sentinelli, Isomorphisms of Hecke modules and parabolic Kazhdan-Lusztig polynomials, J. Algebra 403 (2014), 1–18. MR 3166061Google Scholar
  41. [SW04]
    Zvezdelina Stankova and Julian West, Explicit enumeration of 321, hexagon-avoiding permutations, Discrete Math. 280 (2004), no. 1-3, 165–189. MR 2043806MathSciNetCrossRefGoogle Scholar
  42. [Ten07]
    Bridget Eileen Tenner, Pattern avoidance and the Bruhat order, J. Combin. Theory Ser. A 114 (2007), no. 5, 888–905. MR 2333139MathSciNetCrossRefGoogle Scholar
  43. [War11]
    Gregory S. Warrington, Equivalence classes for the μ-coefficient of Kazhdan-Lusztig polynomials in S n, Exp. Math. 20 (2011), no. 4, 457–466. MR 2859901 (2012j:05460)Google Scholar
  44. [Wes96]
    Julian West, Generating trees and forbidden subsequences, Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994), vol. 157, 1996, pp. 363–374. MR 1417303Google Scholar
  45. [Yun09]
    Zhiwei Yun, Weights of mixed tilting sheaves and geometric Ringel duality, Selecta Math. (N.S.) 14 (2009), no. 2, 299–320. MR 2480718Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael

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