Comparability in Bruhat orders

TL;DR

The paper investigates comparability probabilities in the weak and strong Bruhat orders on the symmetric group, establishing a sharp leading-order decay for weak-order comparability and a subexponential lower bound for strong-order comparability. It leverages the Robinson–Schensted–Knuth correspondence, Plancherel measure, and Greene–Kleitman–Fomin parameters to recast weak-order comparability in terms of linear extensions of permutation posets and to perform precise shape-based averaging. A key technical achievement is a uniform lower bound on an exponent functional Ψ(λ) over all Plancherel shapes, combined with a deterministic peeling argument, which yields the weak-order upper bound. For the strong order, the authors construct large structured permutation families and validate comparability with high probability via the Gale-order criterion and random-walk deviation analysis, producing the first subexponential lower bound in this regime. Taken together, these results close longstanding gaps in previous bounds and illuminate the intricate connections between Bruhat order comparability, permutation statistics, and geometric-combinatorial structures.

Abstract

We determine the sharp asymptotic scale of the probability that two uniformly random permutations are comparable in weak Bruhat order, showing that $\mathbb{P}(σ_1 \preceq_W σ_2)=\exp\Bigl(\bigl(-\tfrac12+o(1)\bigr)\,n\log n\Bigr)$. This significantly improves both of the best known bounds, due to Hammett and Pittel, which placed this probability between $\exp((-1+o(1))n\log n)$ and $\exp(-Θ(n))$. We also improve the best known lower bound for strong Bruhat-order comparability, due to the same authors, by proving a subexponential lower bound. The Bruhat orders are natural partial orders on the symmetric group, appearing in wide-reaching settings including the geometry of flag manifolds, the representation theory of $\mathfrak{S}_{n}$, and the combinatorics of the permutohedron. To analyze weak Bruhat order, we combine classic analytic, tableau-theoretic, and poset-theoretic tools, including the Plancherel measure and the RSK bijection. For strong Bruhat order we construct large families where members are comparable with high probability. Our proof that members are comparable combines the tableau criterion with an associated random-walk-type deviation process.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2: Stirling Stirling1730
• Remark 2.3
• Theorem 2.4
• Proposition 2.5: McDiarmid M89
• Proposition 2.6: Log-sum inequality CT91
• Proposition 2.7
• Definition 2.8