The Algebraic Combinatorics Seminar

Scheduled for Fridays at 3:15 PM

Nantel Bergeron and
Mike Zabrocki

In reverse-chronological order.

Date Speaker Title (click titles for abstract) 3(?) Apr 2015 Sarah Brodsky 27 Feb 2015

In ROOM 332Nantel Bergeron 20 Feb 2015 NO SEMINAR 13 Feb 2015 Christophe Hohlweg (LaCIM 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.)6 Feb 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.30 Jan 2015 Maria Elisa Fernandes (Portugal) 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).23 Jan 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.16 Jan 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.5 Dec 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.21 Nov 2014 Cesar Ceballos More on self avoiding walk 7 Nov 2014

In LIBRARYCesar Ceballos More on self avoiding walk 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 examine and squares. He has some computation done. We can also look at other affine groups BC2 or A1xA1.31 Oct 2014 Mike Zabrocki treat with self avoiding candies 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 (Notes).24 Oct 2014 Mercedes Rosas (Spain) 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.17 Oct 2014

SPECIAL TIME: 12:45Philippe 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.10 Oct 2014 Neal Madras 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.3 Oct 2014 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).26 Sep 2014 Shu Xiao Li On the Saturation conjecture for the structure constant of the immaculate non-symmetric functions. He presented a counter example !!!!12 Sep 2014 Nantel Bergeron What to do this year Some topic I have in mind: (please think of more) ** Some open problem about non-commutative lift of Q-Schur function [Mike, I would love it if you could help us set up a Sage worksheet for computation of those] ** Analogue of saturation conjecture for immaculate (here Cesar and Shu Xiao might lead this one) ** I would still like to look seriously again at Fibolan and fibonomials. (Cesar and Farid) ** Non intersecting path in the lattice of affine Coxeter group (Cesar and Mike)

- Unless otherwise indicated, the seminar meets on Fridays at 3:15 pm in Room 210 of the Fields Institute.
- If you are interested in speaking at the seminar, contact Nantel Bergeron.
- You may also be interested in the Applied Algebra Seminar (Monday afternoons at York University).

** About the seminar. **
Every year we pick a new topics to explore.