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Lusztig's sharp bound for Hecke algebras

A mix of human-written and AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.GR — Group Theory

Contributed by Qiyue Tang ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove Lusztig's sharp boundedness conjecture for Hecke algebras of finite-rank Coxeter systems with positive integer-valued weight functions.

Provenance statement

GPT-6-Pro was used in two rounds of mathematical research and proof development, with requested research windows of 8 hours and 5.5 hours, respectively (13.5 hours in total). GPT-6-Pro was also used for mathematical review, adversarial proof checking, and language editing. An additional audit by Rethlas-OSV will be incorporated upon completion.

References

  1. M. Belolipetsky, Cells and representations of right-angled Coxeter groups, Selecta Math. (N.S.) 10 (2004), no. 3, 325--339. https://doi.org/10.1007/s00029-004-0355-9.DOI
  2. A. Björner and F. Brenti, Combinatorics of Coxeter Groups , Graduate Texts in Mathematics 231 , Springer, New York, 2005.
  3. B. Casselman, Coxeter groups. Part I: Geometry and combinatorics , online notes, https://personal.math.ubc.ca/~cass/coxeter/r.html.link
  4. X. Chen and H. Hu, Boundedness of Lusztig's aa-function for Coxeter groups of finite rank, Invent. Math. (2026), published online 23 September 2026. https://doi.org/10.1007/s00222-026-01453-1; final preprint version, https://arxiv.org/abs/2503.06432v10.DOI
  5. M. J. Dyer, Hecke algebras and reflections in Coxeter groups , Ph.D. thesis, University of Sydney, 1987. https://academicweb.nd.edu/~dyer/papers/dyerthesis.pdf.link
  6. J. Gao, The boundness and lowest two-sided cell of weighted Coxeter groups of rank 33, J. Lie Theory 32 (2022), no. 2, 499--518. https://arxiv.org/abs/1607.02286v10.link
  7. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165--184. https://doi.org/10.1007/BF01390031.DOI
  8. Y. Li and J.-y. Shi, The boundedness of a weighted Coxeter group with non-33-edge-labeling graph, J. Algebra Appl. 18 (2019), no. 5, 1950085, 43 pp.
  9. G. Lusztig, Cells in affine Weyl groups, in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math. 6 , 1985, 255--287. https://doi.org/10.2969/aspm/00610255.DOI
  10. G. Lusztig, Hecke Algebras with Unequal Parameters , CRM Monograph Series 18 , American Mathematical Society, Providence, RI, 2003; revised version, 2014, https://arxiv.org/abs/math/0208154v2.link
Show all 17 referencesShow fewer references
  1. G. Lusztig, Open problems on Iwahori--Hecke algebras, 2020, arXiv:2006.08535. https://arxiv.org/abs/2006.08535.arXiv
  2. J. Parkinson and Y. Yau, Cone types, automata, and regular partitions in Coxeter groups, Adv. Math. 398 (2022), 108146. https://doi.org/10.1016/j.aim.2021.108146.DOI
  3. J.-y. Shi and G. Yang, The weighted universal Coxeter group and some related conjectures of Lusztig, J. Algebra 441 (2015), 678--694. https://doi.org/10.1016/j.jalgebra.2015.05.021.DOI
  4. J.-y. Shi and G. Yang, The boundedness of the weighted Coxeter group with complete graph, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4573--4581. https://doi.org/10.1090/proc/13154.DOI
  5. N. Xi, Representations of Affine Hecke Algebras , Lecture Notes in Mathematics 1587 , Springer-Verlag, Berlin, 1994.
  6. N. Xi, Lusztig's aa-function for Coxeter groups with complete graphs, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 1, 71--90. https://arxiv.org/abs/1106.2424.link
  7. P. Zhou, Lusztig's aa-function for Coxeter groups of rank 33, J. Algebra 384 (2013), 169--193. https://doi.org/10.1016/j.jalgebra.2013.02.029.DOI

Version history

  1. v1 · 19 pagesSubmitted by Qiyue TangInitial depositCurrentNov 08, 2026