UBC Discrete Math Seminar: Lucas Gagnon

A “quasisymmetric variety” (QSV) is a subset of permutations that (when considered as points in n-space) have a vanishing ideal that shares key properties with the ideal generated by quasisymmetric polynomials, and moreover which forms a basis for the Temperley—Lieb algebra TL(2). This talk will give some combinatorial motivation for finding a QSV by telling the story of its discovery and describing some surprisingly nice results found along the way. Of particular interest is a new equivalence relation on permutations defined using their excedance sets. This relation has many desirable properties: each equivalence class is naturally indexed by a noncrossing partition and also forms an interval in the (strong) Bruhat order on permutations, so that the underlying equivalence relation gives a quotient of the Bruhat order. Furthermore, this relation gives a simple method for constructing bases of the Temperley—Lieb algebra, generalizing known results of Williams—Gobet and Zinno. All of this turns out to be the key to solving solving the quasisymmetric variety problem, and I will conclude by connecting these dots: the set of Bruhat-maximal elements from all excedance classes turns out to be a QSV, and the Bruhat and noncrossing partition combinatorics outlined above are essential for the proof. Based on joint work with Nantel Bergeron.