Combinatorial invariance of Kazhdan–Lusztig polynomials

1 paperLean

Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

Combinatorial invariance of Kazhdan–Lusztig polynomials

OpenAI · September 24, 2026

We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.

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Cite

@misc{OAI:Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026,
  author = {{OpenAI}},
  title = {{Combinatorial invariance of Kazhdan--Lusztig polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf}{OAI:Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Combinatorial invariance of Kazhdan–Lusztig polynomials

Scope

The combinatorial invariance conjecture asks whether a Kazhdan–Lusztig polynomial depends only on its Bruhat interval as an ordered set. The formalization proves that every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial. The two intervals may come from different Coxeter systems.

Comparator links

Result	Comparator statement
Combinatorial invariance of Kazhdan–Lusztig polynomials	KLInvariance.lean

Scope note on GitHub →