The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by John Machacek and Nantel Bergeron.

During 2018-19, the seminar takes place from 15:00-16:00 in Ross Building room N638. If you come by public transportation, there is a York University subway station on the TTC Line 1 Yonge-Univerity route. If you come by car, you can find the available parking lots here.

The seminar has been running since 1997. The topics of talks have typically been any mixture of algebra with any other field: combinatorics, geometry, topology, physics, etc. Further down this page you will find links to the seminar webpages for previous years. The audience usually consists of 6–12 people, including several graduate students and post-docs. For this reason, speakers are encouraged to devote a portion of their talk to the suggestion of open problems and the directions for research in their area. If you are interested in speaking at the seminar, contact John Machacek or Nantel Bergeron.You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

Dates are listed in reverse-chronological order. Unless otherwise indicated, all talks will take place on Monday from 15:00-16:00 in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 3 Dec. 2018 Gabriel Frieden

(CRM-ISM)An affine generalization of evacuation The Robinson--Schensted correspondence is a bijection between permutations and pairs of standard Young tableaux of the same shape. Under this bijection, the reverse complement of a permutation corresponds to the evacuation of the two tableaux. The number of standard tableaux of shape \lambda which are fixed by evacuation is equal to f^\lambda(-1), where f^\lambda(q) is the q-analogue of the hook-length formula. Building on work of Shi, Chmutov--Pylyavskyy--Yudovina recently introduced a generalization of Robinson--Schensted which maps elements of the affine symmetric group to pairs of tabloids (= standard row-strict tableaux) of the same shape. There is a natural involution of the affine symmetric group that generalizes the reverse complement of permutations. In this talk, we give an explicit description of the corresponding ``affine evacuation'' map on tabloids, and we show that the number of tabloids fixed by this map is equal to the evaluation of a certain Green's polynomial at q = -1. Along the way, we discover a combinatorial interpretation of the evaluation of the Kostka--Foulkes polynomials at q = -1. This is based on joint work with Mike Chmutov, Dongkwan Kim, Joel Lewis, and Elena Yudovina.26 Nov. 2018

No Seminar19 Nov. 2018 Tianyuan Xu

(Queen's University)On the subregular J-rings of Coxeter systems Let $(W,S)$ be an arbitrary Coxeter system and let $G$ be its Coxeter diagram. We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of $(W,S)$, an associative algebra closely related to the Iwahori--Hecke algebra of $(W,S)$, and present some results on a subalgebra $J_C$ of $J$ that we call the subregular $J$-ring. We show that while products in $J$ are defined in terms of Kazhdan--Lusztig polynomials, they can be computed in $J_C$ by a simple combinatorial algorithm centered around a certain truncated Clebsch--Gordan rule. As applications, we relate $J_C$ to the path algebra of a quiver whose underlying graph is $G$ and deduce some results on the structure and representations of $J_C$. This is joint work with Ivan Dimitrov, Charles Paquette, and David Wehlau.12 Nov. 2018

Pauline Hubert

(UQAM)The bi-characters of spaces of diagonal harmonic polynomials with inert variables The spaces of diagonal harmonic polynomials and some generalizations have been studied a lot since the 1990's. After a quick reminder of already known results, we will introduce a generalization of the space of diagonal harmonics with polynomials containing inert variables. Finnaly we will discuss the computation of the ($GL_r \times S_n$)-character of those spaces.5 Nov. 2018 Aram Dermenjian

(UQAM)Facial weak order in hyperplane arrangements We will discuss a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement. This order is the facial weak order and was first described on Coxeter groups where we showed it to be a lattice in the general case. We provide various characterizations of this poset including a local one, which was first studied by Krob, Latapy, Novelli, Phan, and Schwer in the case of symmetric groups, and a global one, that generalizes the notion of separation sets. These characterizations are the keys to show that for simplicial hyperplane arrangements the facial weak order is a lattice, generalizing a result by Björner, Edelman and Ziegler showing the poset of regions is a lattice for simplicial arrangements.29 Oct. 2018

Robin Sulzgruber

(York University)P-partitions and p-positivity Using the combinatorics of alpha-unimodal permutations we prove two new results on quasi-symmetric functions: (1) an expansion of Gessel's fundamental basis into quasi-symmetric power-sums, and (2) a positive expansion of Stanley's P-partition generating functions into quasi-symmetric power-sums. As a consequence we obtain a unified proof that many interesting families of (quasi-)symmetric functions expand positively into power-sums, for example chromatic quasi-symmetric functions and certain LLT polynomials.22 Oct. 2018 Hugo Mlodecki

(Paris-Sud)Auto-duality of WQSym, the Hopf algebra on packed words A Hopf algebra is a formalism that makes it possible to study algorithms on combinatorial object assembling and disassembling. Among these objects, we find the permutations where each number between 1 and n appears one and only once, or the packed words where each number between 1 and m appears at least once. We will study different ways of assembling and disassembling permutations and the relationships of duality and simple auto-duality that these operations verify. We will then try to generalize these relations to the packed words along a more complex path.15 Oct. 2018 John Machacek

(York University)Locally acyclic cluster algebras and their quivers We will dicuss locally acyclic cluster algebras and techniques for showing a cluster algebra is locally acyclic. We will also explore to realationship between the existence of a reddening sequence and equality of a cluster algebra and its upper cluster algebra.8 Oct. 2018 ThanksgivingNo Seminar1 Oct. 2018 Kelvin Chan

(York University)Induction Relations in the Symmetric Groups and Jucys-Murphy Elements Transitive factorizations faithfully encode many interesting objects. The well-known ones include ramified coverings of the sphere and hypermaps. Enumeration of specific classes of such objects have been known for quite some time now. Hurwitz numbers, monotone Hurwitz numbers and hypermaps numbers were discovered using different techniques. Recently, Carrell and Goulden found a unified algebraic approach to count these objects in genus 0. Jucys-Murphy elements and centrality play important roles in establishing induction relations. Such a method is interesting in its own right. Its corresponding combinatorial decomposition is however intriguingly mysterious. Towards a understanding of direct combinatorial analysis of multiplication of arbitrary permutations, we consider methods, especially operators on symmetric functions, and related problems in symmetric groups.24 Sep. 2018 LaCIM celebrationNo Seminar

Below you will find links to the seminar webpages for previous years.