Title: Partial orders on the symmetric group
| Speaker: | Oliver Pechenik |
| Affiliation: | University of Waterloo |
| Zoom: | Please email Emma Watson |
Abstract:
The symmetric group of permutations is naturally a poset in at least 4 different ways, the (strong) Bruhat order and three flavors of weak order. Stanley showed in 1980 that the Bruhat order is Sperner, essentially meaning that the obvious large antichains are in fact the largest possible. The corresponding fact for weak orders was open until last year, when it was established by Gaetz and Gao. We use partial derivatives and combinatorial tools from Schubert calculus to prove a conjectural refinement due to Stanley. We then use the interplay between the various weak orders to uncover new features of the classical major index statistic on permutations, with applications to the Castelnuovo-Mumford regularity of certain generalized determinantal varieties. (The first part is joint with Zachary Hamaker, David Speyer and Anna Weigandt, and the second part joint with Speyer and Weigandt.)