Combinatorial invariance of Kazhdan–Lusztig polynomials

Follow-up works

• F. Caselli and M. Marietti, A simple classification of the special matchings in the symmetric group (arXiv:2610.21647, 15 Oct 2026). Gives a combinatorial algorithm for Kazhdan–Lusztig polynomials on an isomorphism-closed family of type-A Bruhat intervals; its Remark 1.2 cites this manuscript as [28] and notes that it "claims to prove the Combinatorial Invariance Conjecture".
• Nicolás Libedinsky, Polytopal invariance for Lusztig’s q-weight multiplicities (arXiv:2610.14275v1, 2026-10-09). Cites the OpenAI manuscript as an announced proof, establishes a metric spherical analogue, and proposes in §6.1 a conjecture connecting polytope isometries to Bruhat interval isomorphisms; the AI-use disclosure describes an AI-produced initial argument subsequently corrected and expanded by the author.

Formalization

• OpenAI, Combinatorial invariance of Kazhdan–Lusztig polynomials: Lean scope note (OpenAI math release, 2026-10-06). States that the formalization proves that every order isomorphism between Bruhat intervals in arbitrary, possibly different Coxeter systems preserves the equal-parameter Kazhdan–Lusztig polynomial, and links the comparator statement KLInvariance.lean.
• OpenAI math release, comparator statement KLInvariance.lean (release Lean comparator challenge, 2026). Statement-only file: theorem combinatorial_invariance takes an order isomorphism Interval cs u b ≃o Interval cs' u' b' and concludes klPolynomial cs u b = klPolynomial cs' u' b'.
• OpenAI math release, Invariance.lean (release Lean development, 2026). Contains the proof of KLInvariance.combinatorial_invariance, which imports OAI.RepresentationTheory.KazhdanLusztig.SimultaneousInvariance.

Possibly relevant literature (found by the agent, not checked)

• G. Burrull, N. Libedinsky, and D. Plaza, Combinatorial invariance conjecture for $\widetilde{A}_2$ (arXiv:2105.04609, 2021; IMRN, doi:10.1093/imrn/rnac105). Proves the conjecture for the affine Weyl group $\widetilde{A}_2$, a previously known infinite case of the statement the paper says includes infinite groups.
• G. T. Barkley and C. Gaetz, On combinatorial invariance of parabolic Kazhdan–Lusztig polynomials (arXiv:2404.04246, 2024; Selecta Math. 31 (2025), 51). Shows that the Lusztig–Dyer conjecture is equivalent to Marietti's parabolic analogue and a refined parabolic version, which bears on what Theorem 1.1 would imply in the parabolic setting.
• F. Esposito and M. Marietti, Flipclasses and Combinatorial Invariance for Kazhdan–Lusztig polynomials (arXiv:2402.13097, 2024). Introduces the flipclass approach that the introduction cites only through the later Esposito–Marietti–Stella paper.
• F. Esposito, M. Marietti, G. T. Barkley, and C. Gaetz, A note on Combinatorial Invariance of Kazhdan–Lusztig polynomials (arXiv:2404.12834, 2024). Formulates a hypercube-decomposition conjecture that implies type-A combinatorial invariance, related to the hypercube approach surveyed in the introduction.