Please join the Department of Mathematics and Statistics on Monday, September 27th at 3:30pm in Votey Hall 209 for a fascinating combinatorics seminar featuring Harvard University's, NSF Postdoctoral Fellow, Christian Gaetz.
Linear Extensions, the 1/3-2/3 Conjecture, and Coxeter Groups
NSF Postdoctoral Fellow Christian Gaetz, Harvard University
Monday, September 27th, 3:30 PM, Votey Hall 209
Abstract: Given partial information about a totally ordered set, it is a natural problem to try and find a pair x,y of elements whose comparison will give us as much additional information about the underlying total order as possible. Phrased in terms of linear extensions of posets, the "1/3-2/3 Conjecture", originally formulated in 1968, gives a precise bound on how well this can be done. After some background and survey of known partial results, I will explain how this problem can be reinterpreted in terms of convex subsets of the symmetric group, and thereby generalized to convex subsets of any Coxeter group. Remarkably, we conjecture that the 1/3-2/3 Conjecture 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 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. Joint work with Yibo Gao.