Bruhat intervals that are large hypercubes

TL;DR

The paper identifies unusually large Bruhat intervals in $S_n$ that are poset hypercubes, by introducing and characterizing dyadically well-distributed permutations $\mathcal{D}_m$. It proves that for $n=2^m$, the interval $[x_m,y_m]$ with $x_m,y_m\in\mathcal{D}_m$ is a hypercube of dimension $m2^{m-1}$, via an explicit poset isomorphism to $S_2^{\mathcal{C}_m}$. This construction yields a tight, near-maximal hypercube size and provides exact $d$-invariant values $d_{x_m,y_m}=m2^{m-1}$, linking to the Kazhdan-Lusztig theory and to cluster algebras through the equality of frozen variables and $d$-invariants. The work further connects to low-discrepancy sequences, generalizes to $t$-adic settings, and offers insights into open Richardson varieties and moduli of Bruhat-graph embeddings. Overall, it demonstrates both a novel mathematical structure and a path from AI-assisted discovery to rigorous combinatorial and geometric conclusions, with implications for associated algebraic and geometric frameworks.

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.

Figures (5)

• Figure 1: The first "unexpected" hypercube in $S_4$ of dimension $4$. This hypercube is also the first interesting example of our construction.
• Figure 2: $M_{x_4}$ (blue) and $M_{y_4}$ (red) with fundamental blocks highlighted.
• Figure 3: Vertices in a fundamental block
• Figure 4: Dividing a complementary block as in the proof of Lemma \ref{['lem:order-preserving']}
• Figure 5: Rank two subgraphs of Bruhat intervals for symmetric groups: the arrow, incomplete diamonds, diamond and full $S_3$.

Theorems & Definitions (35)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• proof