Flipclasses and Combinatorial Invariance for Kazhdan–Lusztig polynomials

• Francesco Esposito ^[1] ; Mario Marietti ^[2]
  1. [1] University of Padua

     University of Padua

     Padova, Italia

     
  2. [2] Marche Polytechnic University

     Marche Polytechnic University

     Ancona, Italia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01099-6
• Enlaces
  • Texto completo
• Resumen

  • In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan–Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of qh of the Kazhdan–Lusztig R -polynomials, for h ≤ 6. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.

• Referencias bibliográficas
  • 1. Barkley, G.T., Gaetz, C.: Combinatorial invariance for Kazhdan–Lusztig R-polynomials of elementary intervals, Math. Ann. (2025). https://doi.org/10.1007/s00208-025-03175-w
  • 2. Billera, L., Brenti, F.: Quasisymmetric functions and Kazhdan-Lusztig polynomials. Israel J. Math. 184, 317–348 (2011)
  • 3. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231. Springer-Verlag, New York (2005)
  • 4. Blanco, S.: The complete cd-index of dihedral and universal Coxeter groups. Electron. J. Combin. 18, 174, 16 (2011)
  • 5. Blundell, C., Buesing, L., Davies, A., Veli˘ckovi´c, P.,Williamson, GWilliamson: Towards combinatorial invariance for Kazhdan-Lusztig polynomials....
  • 6. Brenti, F.: A combinatorial formula for Kazhdan-Lusztig polynomials. Invent. Math. 118, 371–394 (1994)
  • 7. Brenti, F.: Some open problems on Coxeter groups and unimodality, In: Berkesch, C., Brubaker, B., Musiker, G., Pylyavskyy, P., Reiner,...
  • 8. Brenti, F., Caselli, F., Marietti, M.: Special Matchings and Kazhdan-Lusztig polynomials. Adv. Math. 202, 555–601 (2006)
  • 9. Brenti, F., Caselli, F., Marietti, M.: Diamonds and Hecke algebra representations. Int. Math. Res. Not. IMRN 2006, 29407 (2006)
  • 10. Brenti, F., Incitti, F.: Lattice paths, lexicographic correspondence and Kazhdan-Lusztig polynomials. J. Algebra 303, 742–762 (2006)
  • 11. Brenti, F., Marietti, M.: Kazhdan-Lusztig R-polynomials, combinatorial invariance, and hypercube decompositions. Math. Z. 309, 25 (2025)
  • 12. Burrull, G., Libedinsky, N., Plaza, D.: Combinatorial invariance conjecture for A˜2. Int. Math. Res. Not. IMRN 2023(10), 8903–8933 (2023)
  • 13. Caselli, F., D’Adderio, M., Marietti, M.: Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids. Int. Math. Res. Not....
  • 14. Davies, A., P. Veli˘ckovi´c, Buesing, L., Blackwell, S., Zheng, D Zheng, Tomašev, N., Tanburn, R., Battaglia, P., Blundell, C., Juhász,...
  • 15. Dyer, M. J." Hecke algebras and reflections in Coxeter groups, Ph. D. Thesis, University of Sydney, (1987)
  • 16. Dyer, M.J.: On the Bruhat graph”of a Coxeter system. Compos. Math. 78, 185–191 (1991)
  • 17. Dyer, M.J.: Hecke algebras and shellings of Bruhat intervals. Compos. Math. 89, 91–115 (1993)
  • 18. Esposito, F., Marietti, M.: A note on combinatorial invariance of Kazhdan–Lusztig polynomials (with an appendix by G.T. Barkley and C....
  • 19. Gurevich, M., Wang, C.: Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition. Sel. Math. New Ser. 30,...
  • 20. Hultman, A.: Bruhat intervals of length 4 in Weyl groups. J. Combin. Theory Ser. A 102, 163–178 (2003)
  • 21. Incitti, F.: More on the combinatorial invariance of Kazhdan-Lusztig polynomials. J. Combin. Theory Ser. A 114, 461–482 (2007)
  • 22. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)
  • 23. Marietti, M.: Algebraic and combinatorial properties of zircons. J. Algebraic Combin. 26, 363–382 (2007)
  • 24. Marietti, M.: Special matchings and parabolic Kazhdan-Lusztig polynomials. Trans. Amer. Math. Soc. 368(7), 5247–5269 (2016)
  • 25. Marietti, M.: The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals. Advances in Math....
  • 26. The Sage Developers, SageMath, the Sage Mathematics Software System (Version 10.2), (2023), http:// www.sagemath.org
  • 27. Tits, J.: Le problème des mots dans les groupes de Coxeter, in: Sympos. Math., vol. 1, INDAM, Roma, 175–185 (1969)