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Math-427-A2 · revision r1 · on Math-427

OpenAI claim summary for Math-427

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Kazhdan--Lusztig invariance: A directed Bruhat-graph isomorphism preserves reachability and hence the Bruhat-interval order. The claimed poset-isomorphism theorem uses strictly less data and applies to all Coxeter systems.

The claimed result implies the stated question and also supplies the additional strength described above.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Combinatorial Invariance of Kazhdan–Lusztig Polynomials
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Combinatorial Invariance of Kazhdan–Lusztig Polynomials

OpenAI's claim

Kazhdan--Lusztig invariance: The manuscript claims equality of equal-parameter Kazhdan--Lusztig polynomials whenever two Bruhat intervals are isomorphic as posets. The intervals may lie in unrelated Coxeter systems.

Mathematical statement of the claimed result

Mathematical claim 1.
For arbitrary Coxeter systems $(W,S)$ and $(W^\prime,S^\prime)$, choose $u\le b$ and $u^\prime\le b^\prime$ in their Bruhat orders. If the finite interval posets $[u,b]$ and $[u^\prime,b^\prime]$ are abstractly isomorphic, then the claimed equal-parameter Kazhdan--Lusztig equality is \[P^W_{u,b}(q)=P^{W^\prime}_{u^\prime,b^\prime}(q).\] Neither the ambient Coxeter type nor a preservation of edge labels by the poset isomorphism is required.

Outline of the argument

Kazhdan--Lusztig invariance: Moment-graph sheaves encode the polynomials as graded ranks of vertex modules. A reflection order from the second interval supplies a transported edge schedule on the first. Induction proves both polynomial equality and freeness of kernels for terminal edge sets; the local edge step makes the sheaf calculation independent of the original labels.

Formal verification

Family 168 describes Lean coverage for preservation of equal-parameter polynomials under Bruhat-poset isomorphisms across Coxeter systems.

Kazhdan--Lusztig invariance: The manuscript links KLInvariance.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources