The 1/3-2/3 Conjecture for Coxeter groups
报告人：高奕博（麻省理工学院）
时间：2022年5月27日9:00-10:00
地点：腾讯会议：967-229-265
报告摘要：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 Coxeter 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 (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. This is joint work with Christian Gaetz.
报告人简介：高奕博，2022年博士毕业于麻省理工大学，将前往北京国际数学研究中心任助理教授，主要从事Coxeter群和Schubert计数演算的研究。