Year: 2026
Type: article
Abstract: We study the question of finding big Bruhat intervals that are poset hypercubes in the symmetric group $S_{n}$. Using permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we were led to an unusual situation in which the agent produced a pattern that performed well for the $n$ tested, and that we show works well for general $n$. When $n$ is a power of 2 we exhibit a hypercube of dimension $O(n\log n)$, matching the largest possible dimension up to a constant multiple. Furthermore, we give an exact characterization of the vertices of this hypercube: they are precisely the dyadically well-distributed permutations—a simple digitwise property that already appeared in connection with Monte Carlo integration and mathematical finance. The maximal dimension of a Bruhat interval that is a hypercube in $S_{n}$ gives a lower bound (and possibly is equal to) the maximal possible coefficient of the second-highest degree term in the Kazhdan–Lusztig $R$-polynomial in $S_{n}$. As a surprising consequence, we obtain a new lower bound of order $n\log n$ for the maximal number of frozen variables appearing in the cluster algebras attached to the open Richardson varieties in $S_{n}$, and a similar result for moduli spaces of embeddings of Bruhat graphs.
DOI: 10.1093/imrn/rnag222
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7118458454
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