SCIENTIFIC PROGRAMS AND ACTIVITIES

Nantel Bergeron

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.)

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.

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).

Farid Aliniaeifard
More on the problem of Fibo-Catalans

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

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.

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.

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.

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.

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.

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.

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).