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) 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.