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Large Hypercube Intervals in Bruhat Order

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bruhat-hypercube-intervalsCombinatoricsposer not recordedrecorded: partial

1 attempt · no person has looked

Statement

How large can a Bruhat interval in SnS_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(nlog⁡n)O(n \log n) for nn a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.

Context

Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.

A question in the active problem circle around combinatorial invariance of Kazhdan-Lusztig polynomials, pursued by leading figures of the field.

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  • #1

    Attempt 1

    constructionAlphaEvolve with Jordan Ellenberg, Nicolas Libedinsky, David Plaza, José Simental, Geordie Williamson ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    AlphaEvolve
    people
    Jordan Ellenberg, Nicolas Libedinsky, David Plaza, José Simental, Geordie Williamson

    AlphaEvolve, searching evolutionarily rather than exhaustively, "produced a pattern which performed well for the n tested, and which we show works well for general n" - the agent found the construction, the humans proved it works in general.

    Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.

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