BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yibo Gao (MIT) DTSTART:20201019T190000Z DTEND:20201019T200000Z DTSTAMP:20260423T022744Z UID:YUAAS/3 DESCRIPTION:Title: Th e 1/3-2/3 Conjecture for Coxeter groups\nby Yibo Gao (MIT) as part of York University Applied Algebra Seminar\n\n\nAbstract\nThe 1/3-2/3 Conject ure\, originally formulated in 1968\, is one of the best-known open proble ms 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 o f 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 Cox eter group\, with new and interesting equality cases appearing. We general ize 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 o f the case of width-two posets)\, we give a uniform lower bound for balanc e constants in all finite Weyl groups using a new generalization of order polytopes to this context\, and we introduce generalized semiorders for wh ich 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 Conjec ture\, and therefore on which methods are likely to be successful in resol ving it. This is joint work with Christian Gaetz.\n LOCATION:https://researchseminars.org/talk/YUAAS/3/ END:VEVENT END:VCALENDAR