University of Minnesota Combinatorics Seminar
Friday, January 27, 2011
3:35pm in 570 Vincent Hall



Antichains in minuscule posets

Vic Reiner

University of Minnesota


Abstract

Minuscule posets arise in representation theory of simple Lie groups: they are the posets whose distributive lattice of order ideals indexes the weights in a representation having only one Weyl group orbit of weights. We will begin by reviewing the magical enumerative properties of minuscule posets, known from work of Proctor, Stanley, and Stembridge.

Then we will discuss further minuscule magic uncovered in the 2011 Univ. of Minnesota REU work of MIT undergraduates David B. Rush and Danny Shi. They proved a conjecture predicting the orbit structure for the Duchet-Brouwer-Schrijver-FonDerFlaass action on antichains in a minuscule poset, as well as antichains in its product with a two-element chain. Both results proceed via root-of-unity evaluations of the generating function counting the order ideals by size.

(Based on their preprint-- arXiv:1108.5245).