SCIENTIFIC PROGRAMS AND ACTIVITIES

March 28, 2024
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
Algebraic Combinatorics Seminar 2014-15
at the Fields Institute

Fridays 3:15 - 4:30 p.m.

Organizer: Nantel Bergeron and Tom Denton
(York University)

OVERVIEW

The purpose of this seminar is to cover exposition on topics of algebraic combinatorics which are of interest to the people attending, so please feel free to come and participate. Every year we pick a new topic to explore. We will be selecting the seminar topic for this year shortly, so attend the first few talks if you want to influence the decision.

We also organize special sessions jointly with the Applied Algebra Seminar (York University).

Important: Seminars during the month of May will be bmoved to the Bahen Center, room BA 1220

 

Fridays, 3:15-4:30pm Upcoming Seminars at the Fields Institute, Room 210
March 27, 2015

Nantel Bergeron

March 30, 2015
Hugh Thomas
April 1, 2015 Sarah Brodsky
Past Seminars
Feb 13, 2015

Christophe Hohlweg (UQAM)
Small roots, low elements and weak order in Coxeter groups

Let (W,S) be a Coxeter system: Is the smallest subset of W containing S, closed under join (for the right weak order) and suffix finite? In this talk we will explain that this question, which arose in the context of Artin-Tits Braid groups, has an affirmative answer. The proof reveals nice connections between the weak order, the Bruhat order, inversion sets and small roots. Small roots are the main ingredient introduced by Brink and Howlett in order to build a `canonical automatonâ that recognizes the language of reduced words in a Coxeter group. From small roots and inversion sets, we define a new finite class of elements in W called `low elementsâ that answer the question. Low elements seem rich in further applications in the study of infinite Coxeter groups, which will discuss if time permits. (Based on joint works with Patrick Dehornoy and Matthew Dyer.)

Feb 20, 2015 NO SEMINAR
Feb 27, 2015
Room 332
Nantel Bergeron
Feb 6, 2015

Dimitri Leemans (New Zealand)
Abstract polytopes and projective lines

We will discuss the classification of abstract polytopes whose automorphism group is an almost simple group of PSL(2,q) type. We will detail the classification of the regular polytopes that we started together with Egon Schulte for the groups PSL(2,q) and PGL(2,q) and that we later extended with Thomas Connor and Julie De Saedeleer. We will also explain where we stand, together with Eugenia O'Reilly-Regueiro and Jeremie Moerenhout, for the chiral polytopes related to these groups.

Jan 30, 2015

Maria Elisa Fernandes
Spherical and toiroidal hypertopes

Hypertopes are thin residually connected geometries. An hypertope is regular if it is ag transitive and is chiral if it has two orbits on the ags. Abstract regular polytopes are examples of hypertopes, those with linear Coxeter Diagram. One of our focus is the classication of hypertopes of a certain type. Here we consider spherical, locally spherical and locally toroidal hypertopes (hypertopes having all parabolic subgroups either spherical or toroidal).

Jan 23, 2015

Farid Aliniaeifard
More on the problem of Fibo-Catalans

Farid will continue on the recent progress we have regarding the q-Fibo-Catalans polynomials.

Jan. 16, 2015

Shu Xiao Li
Solving local equations for self avoiding walks on affine type B lattice

Shu Xiao present his solution of the local equations we need to understand the self avoiding walk on the affine type B lattice. He also discuss what will go wrong in the rest of the proof and what we need.

Dec. 5, 2014

Farid Aliniaeifard
On the problem of Fibo-Catalans

Farid will recall what we have done so far with the problem of showing the positivity of the q-Fibo-Catalans polynomials. Will go up to the recent progress we made on it.

Nov. 28, 2014 working seminar
Nov. 21, 2014

Cesar Ceballos
More on self avoiding walks

We will continue to work on self-avoiding walk problems on lattices of affine Coxeter groups. Mike suggested last week to look at Affine G2 which look like a thick hexagonal lattice filled with exagone and squares. We have started to set equations. We plan to write down the equations and see if there is a solution.

Nov. 7, 2014
Stewart Library
Cesar Ceballos and Mike Zabrocki
More on self avoiding walks
Oct 31, 2014

Mike Zabrocki
More on self avoiding walks

I will describe an algebraic way of looking at self avoiding walks and concentrate on these walks in the group lattice of reflection groups. I will demonstrate how to compute the numbers of self avoiding walks using non-commutative Grobner bases in GAP and Sage.

Oct. 24, 2014 Mercedes Rosas
Symmetries for the structural constants for the ring of symmetric functions.

We describe a family of closely related symmetries that share the main structural constants for symmetric functions. This includes the Littlewood Richardson, the Kronecker, and the plethysm coefficients, among others.

October 17
Stewart Library

12:45pm-2:00pm

Philippe Nadeau (France)
Combinatorics of the affine Temperley-Lieb algebra

The classical Temperley-Lieb algebra was originally defined in statistical mechanics, but has since come up in numerous branches of mathematics, such as knot theory or representation theory. It possesses a well known faithful representation as an algebra of noncrossing diagrams, with a basis naturally indexed by 321-avoiding permutations. Such combinatorics generalize naturally to a certain affine version of the Temperley-Lieb algebra, and I will describe several combinatorial aspects of this affine setting.

October 10

Neal Madras (York University, Fields Institute)
Self-Avoiding Walks on the Hexagonal Lattice

A self-avoiding walk in a lattice is a path that does not intersect itself. The number of n-step self-avoiding walks starting at the origin is approximately C^n for some constant C that depends on the lattice. The exact value of C is rarely known exactly. An exception is the hexagonal lattice, where H. Duminil-Copin and S. Smirnov proved that C = sqrt{ 2 + sqrt{2} } (Annals of Mathematics 175, 1653-1665, 2012, arXiv:1007.0575), verifying a 3-decade-old physics prediction. I will review their proof. As for extending their method to other lattices, the primary obstacle seems to be geometric for three-dimensional lattices, but algebraic for other planar lattices.

October 3

Christophe Hohlweg (LaCIM, UQAM)
Weak order and imaginary cone in infinite Coxeter groups

The weak order is a nice combinatorial tool intimately related to the study of reduced words in Coxeter groups. In this talk, we will discuss a conjecture of Matthew Dyer that proposes a generalization of the framework weak order/reduced words to infinite Coxeter groups. On the way, we will talk of the relationships between limits of roots and tilings of their convex hull, imaginary cones, biclosed sets and inversion sets of reduced infinite words (partially based on joint works with M. Dyer, J.P. Labbé and V. Ripoll).

September 26 Shu Xiao Li
A tentative proof of the Saturation conjecture for the structure constant of the immaculate non-symmetric functions.
September 12
September 19 (No Seminar)