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Large Hypercube Intervals in Bruhat Order

Combinatorics · partial

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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.

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  1. construction · #1

    Jordan Ellenberg, Nicolas Libedinsky, David Plaza, José Simental and Geordie Williamson, using AlphaEvolve

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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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