Combinatorial invariance of Kazhdan–Lusztig polynomials
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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I'll remark on some problems in algebraic combinatorics. Combinatorial invariance of KL polynomials . Proved true, I'd say in agreement with the majority's sentiment. Shareshian--Wachs conjecture. Proved true, in line with nearly universal expectations. Saxl's conjecture. Proved true, in line with…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
says that isomorphic Bruhat intervals have the same equal-parameter Kazhdan–Lusztig polynomial, even when the intervals come from different Coxeter systems. Ramanujan graphs and higher-dimensional expanders. The subject of New Ramanujan Graphs! returns with a deterministic polynomial-time…