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The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials

Theorem 1.1: for arbitrary Coxeter systems (W,S)(W,S) and (W′,S′)(W',S'), including infinite and noncrystallographic ones, and u≤bu\le b, u′≤b′u'\le b', any poset isomorphism [u,b]→[u′,b′][u,b]\to[u',b'] gives Pu,bW=Pu′,b′W′P^W_{u,b}=P^{W'}_{u',b'} for the equal-parameter polynomials; restricting to subintervals determines every Px,yP_{x,y} and, by reciprocity, every RR-polynomial on the interval. The proof uses moment-graph sheaves (Braden-MacPherson, Fiebig, Elias-Williamson), an edge schedule transported from a reflection order of the second system, and Dyer's path formula. It gives no procedure computing PP from the poset, and it treats equal parameters only.

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Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials · KL combinatorial invariance

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Lean: OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean (combinatorial_invariance)

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Theorem 1.1: for arbitrary Coxeter systems (W,S)(W,S) and (W′,S′)(W',S'), including infinite and noncrystallographic ones, and u≤bu\le b, u′≤b′u'\le b', any poset isomorphism [u,b]→[u′,b′][u,b]\to[u',b'] gives Pu,bW=Pu′,b′W′P^W_{u,b}=P^{W'}_{u',b'} for the equal-parameter polynomials; restricting to subintervals determines every Px,yP_{x,y} and, by reciprocity, every RR-polynomial on the interval. The proof uses moment-graph sheaves (Braden-MacPherson, Fiebig, Elias-Williamson), an edge schedule transported from a reflection order of the second system, and Dyer's path formula. It gives no procedure computing PP from the poset, and it treats equal parameters only. parent of this event

The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials parent of this event

The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials evidence for this event

Dyer, On the Bruhat graph of a Coxeter system (Compositio Math. 1991) evidence for this event

VibeMathed record: The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials evidence for this event

Lean: OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean (combinatorial_invariance) evidence for this event

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The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials — Mathematical Frontier Network