Algebraic Combinatorics Seminar - Lucas Gagnon

Title: Quasisymmetric varieties, excedances, and bases for the Temperley--Lieb algebra

Abstract:  This talk is about finding a quasisymmetric variety (QSV): a subset of permutations which (i) is a basis for the Temperley--Lieb algebra TL_n(2), and (ii) has a vanishing ideal (as points in n-space) that behaves similarly to the ideal generated by quasisymmetric polynomials.   While this problem is primarily motivated by classical (co-)invariant theory and generalizations thereof, the course of our investigation uncovered a number of remarkable combinatorial properties related to our QSV, and I will survey these as well.  Of particular interest is a new equivalence relation on permutations defined using their excedance sets.  This relation has many nice properties: each equivalence class is naturally indexed by a noncrossing partition and also forms an interval in the (strong) Bruhat order.  This allows us to define a quotient Bruhat order and gives a simple method for constructing many new bases of TL_n(2), generalizing known results of Williams--Gobet and Zinno.  Surprisingly, the combinatorics of this equivalence relation turn out to be key in solving the QSV problem: collecting the maximal element of each excedance class produces a QSV, and the ensuing noncrossing partition combinatorics are essential to prove this fact.  Based on joint work with Nantel Bergeron; arXiv:2302.10814.