Balance constants for Coxeter groups

TL;DR

This paper extends the study of the 1/3-2/3 Conjecture from posets to convex subsets of Coxeter groups, proposing new bounds and cases where the conjecture holds, thus broadening the understanding of this open problem.

Contribution

It generalizes the 1/3-2/3 Conjecture to Coxeter groups, proves it for certain cases, and introduces new tools like generalized semiorders and order polytopes.

Findings

Conjecture holds for convex subsets below fully commutative elements in acyclic Coxeter groups.

Established a uniform lower bound for balance constants in finite Weyl groups.

Resolved the conjecture for generalized semiorders.

Abstract

The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.