Séminaire Lotharingien de Combinatoire, 85B.41 (2021), 12 pp.

Christian Gaetz and Yibo Gao

The 1/3-2/3 Conjecture for Coxeter Groups

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 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 establish our conjecture when C is a weak order interval below a fully commutative element in any acyclic Coxeter group (a 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.

See arXiv version for a full version of this work.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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