On combinatorial invariance of parabolic Kazhdan–Lusztig polynomials

• Grant T. Barkley ^[1] ; Christian Gaetz ^[2]
  1. [1] Harvard University

     Harvard University

     City of Cambridge, Estados Unidos

     
  2. [2] University of California System

     University of California System

     Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 3, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01052-7
• Enlaces
  • Texto completo
• Resumen

  • We show that the Combinatorial Invariance Conjecture for Kazhdan–Lusztig polynomials due to Lusztig and to Dyer, its parabolic analog due to Marietti, and a refined parabolic version that we introduce, are equivalent. We use this to give a new proof of Marietti’s conjecture in the case of lower Bruhat intervals and to prove several new cases of the parabolic conjectures.

• Referencias bibliográficas
  • Barkley, G. T., Gaetz, C.: Combinatorial invariance for Kazhdan-Lusztig R-polynomials of elementary intervals. Math. Ann. (2025)
  • Be˘ılinson, A., Bernstein, J.: Localisation de g-modules. C. R. Acad. Sci. Paris Sér. I Math., 292(1):15– 18, (1981)
  • Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
  • Blundell, C., Buesing, L., Davies, A., Veliˇckovi´c, P., Williamson, G.: Towards combinatorial invariance for Kazhdan-Lusztig polynomials....
  • Boe, B.D.: Kazhdan-Lusztig polynomials for Hermitian symmetric spaces. Trans. Amer. Math. Soc. 309(1), 279–294 (1988)
  • Brenti, F.: A combinatorial formula for Kazhdan-Lusztig polynomials. Invent. Math. 118(2), 371–394 (1994)
  • Brenti, F.: Parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. Trans. Amer. Math. Soc. 361(4), 1703–1729 (2009)
  • Brenti, F., Caselli, F., Marietti, M.: Special matchings and Kazhdan-Lusztig polynomials. Adv. Math. 202(2), 555–601 (2006)
  • Brenti, F., Mongelli, P., Sentinelli, P.: Parabolic Kazhdan-Lusztig polynomials for quasi-minuscule quotients. Adv. in Appl. Math. 78, 27–55...
  • Brylinski, J.-L., Kashiwara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math. 64(3), 387–410 (1981)
  • Burrull, G., Libedinsky, N., Plaza, D.: Combinatorial invariance conjecture for A 2. Int. Math. Res. Not, IMRN (2022)
  • Deodhar, V. V: On some geometric aspects of Bruhat orderings. II. The parabolic analogue of KazhdanLusztig polynomials. J. Algebra, 111(2):483–506,...
  • Dyer, M.: Hecke algebras and reflections in Coxeter groups. PhD thesis, University of Sydney, (1987)
  • Dyer, M.: On the “Bruhat graph” of a Coxeter system. Compositio Math. 78(2), 185–191 (1991)
  • Incitti, F.: More on the combinatorial invariance of Kazhdan-Lusztig polynomials. J. Combin. Theory Ser. A 114(3), 461–482 (2007)
  • Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)
  • Lascoux, A., Schützenberger, M.-P.: Polynômes de Kazhdan & Lusztig pour les grassmanniennes. In: Young tableaux and Schur functors in...
  • Marietti, M.: Special matchings and parabolic Kazhdan-Lusztig polynomials. Trans. Amer. Math. Soc. 368(7), 5247–5269 (2016)
  • Marietti, M.: The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals. Adv. Math. 335, 180–210...
  • Marietti, M.: Bruhat intervals and parabolic cosets in arbitrary Coxeter groups. J. Algebra 614, 1–4 (2023)
  • Sentinelli, P.: Special idempotents and projections. Semigroup Forum 103(1), 261–277 (2021)