Bruhat intervals that are large hypercubes

TL;DR

This paper investigates large hypercube-shaped Bruhat intervals within the symmetric group, revealing a construction that achieves near-maximal dimension and characterizing their vertices through a digitwise property linked to various mathematical fields.

Contribution

It provides a new construction of large Bruhat interval hypercubes in $S_n$, characterizes their vertices as dyadically well-distributed permutations, and connects these findings to Kazhdan--Lusztig polynomials and cluster algebras.

Findings

Constructed hypercubes of dimension $O(n ext{log} n)$ in $S_n$ for $n$ a power of 2.

Characterized vertices of these hypercubes as dyadically well-distributed permutations.

Established lower bounds for cluster algebra variables and moduli spaces based on hypercube dimensions.

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 which performed well for the $n$ tested, and which 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 \emph{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 an 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.