Result 168, Combinatorics

Combinatorial invariance of Kazhdan–Lusztig polynomials

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.

Lean formalization Proof

The bigger picture

Why it matters

A polynomial attached to an ordered interval may seem to depend on the larger algebraic system around it. The manuscript claims that, for equal-parameter Kazhdan-Lusztig polynomials, the interval's order structure alone is enough.

What changes?

Coxeter systems describe groups using generators that behave like reflections. Bruhat order is a partial ordering of their elements, and a Bruhat interval consists of all elements between two endpoints. Those endpoints also determine a Kazhdan-Lusztig polynomial. The manuscript reports that any order-preserving correspondence between intervals, with an order-preserving inverse, forces their equal-parameter polynomials to agree. The claim covers arbitrary Coxeter systems, including comparisons across different systems, and resolves the full combinatorial invariance conjecture in this equal-parameter setting.

What does that help mathematicians do?

If established, this lets researchers transfer a known polynomial to any interval with the same abstract order, without requiring the surrounding Coxeter systems to match. It also gives an obstruction: intervals with different polynomials cannot be isomorphic as ordered sets. The converse does not follow; matching polynomials alone need not establish that two intervals have the same order structure.

Are there practical applications?

The immediate value is foundational: the claimed theorem identifies which information determines these polynomials and which surrounding algebraic context can be discarded. It supports organizing and comparing polynomial calculations by interval structure rather than by their original Coxeter systems. The supplied abstract does not provide a faster computation method or a practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Combinatorial invariance of Kazhdan–Lusztig polynomials

September 24, 2026 31 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/168.md.

Combinatorial invariance of Kazhdan–Lusztig polynomials

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

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

Posts about this result

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.