The Applied Algebra Seminar

A Monday afternoon research seminar

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 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 at 3:00pm in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 12 Apr. 2010 Ada Chan Type II matrices TBA05 Apr. 2010 Sandeep Bhargava Finite-dimensional representations of ${\mathfrak{g}_{2n,\rho}}( \mathbb{C}_q[t_1^{\pm 1}, t_2^{\pm 1}] )$ An important contribution made by Canadian mathematicians to Lie theory in the last decade and a half involves their work on extended affine Lie algebras, which were first introduced by the physicists Høegh-Krohn and Torrésani in their work on quantum gauge theory.We examine the finite-dimensional irreducible representations of the Lie algebra ${\mathfrak{g}_{2n,\rho}}( \mathbb{C}_q[t_1^{\pm 1}, t_2^{\pm 1}] )$, where $\rho = \pm 1$ and $\mathbb{C}_q[t_1^{\pm 1}, t_2^{\pm 1}]$ is the associative algebra of non-commuting Laurent polynomials over $\mathbb{C}$ with $t_2t_1 = qt_1t_2$ with $q = \pm 1$. (When $\rho=-1$ and $n \geq 2$, this Lie algebra is an example of an extended affine Lie algebra of type $C_n$.)

This is joint work with Hongjia Chen and Professor Yun Gao.

29 Mar. 2010 Alexei Miasnikov Limits of groups, Krull dimension, and Cantor Bendixon rank TBA22 Mar. 2010 TBA15 Mar. 2010 Mahshid Atapour Ratio limit theorem and shape results for pattern-avoiding permutations A permutation $p$ of $\{1,2,...,N\}$ is said to contain a pattern (relatively shorter permutation) $q$ of length $k$ ($k<N$) if $p$ contains a substring of length $k$ that has the same relative order as $q$. Let $S_N(q)$ denote the set of permutations of length $N$ which avoid the pattern $q$. In this talk, I will present a brief sketch of the proof of a ratio limit theorem for the number of $q$-avoiding permutations when $q$ belongs to some specific classes. Considering a permutaion of length $N$ as a set of $N$ points in the $xy$-plane, I will also discuss some results about the typical shape of some $q$-avoiding permutations in the $xy$-plane.This is joint work with my postdoc supervisor Neal Madras.

12 Mar. 2010

*special day*Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics The cyclic sieving phenomenon (CSP) was introduced in 2004 by Reiner, Stanton and White and generalizes Stembridge's $q=-1$ phenomenon. It appears in various contexts and in particular in Coxeter-Catalan combinatorics: for example, several instances of the CSP can be found in the context of non-crossing partitions associated to Coxeter groups. I will define the CSP in general and will give several examples. Moreover, I will introduce a less known instance of the CSP on non-crossing partitions using the Kreweras complement and will relate it to a new instance on non-nesting partitions which can be associated to crystallographic Coxeter groups.8 Mar. 2010 Alexander Yong Drift configurations I'll definedrift configurationsand use them to give compatible and new combinatorial rules for Kazhdan-Lusztig polynomials and (Hilbert-Samuel) multiplicities of vexillary Schubert varieties. This is joint work with Li Li (U. Illinois at Urbana-Champaign).1 Mar. 2010 Hugh Thomas Littlewood-Richardson rules for K-theory I will review the combinatorics of increasing tableaux, as introduced by Alex Yong and me, and recall how this combinatorics can be used to give a formula for the structure constants for the K-theory of Grassmannians. I will then explain how shifted increasing tableaux can be used similarly to provide a formula for the structure constants of the K-theory of odd orthogonal Grassmannians. An essential ingredient in our proof is the Pieri rule for K-theory of odd orthogonal Grassmannians, recently obtained by Buch and Ravikumar. The other ingredients are essentially tableau combinatorics; they will be the focus of my talk. The work to be described is all joint with Alex Yong, and is contained in the preprints arXiv:1002.1664 (for odd orthogonal Grassmannians) and arXiv:0705.2915 (for usual Grassmannians).22 Feb. 2010 Mike Zabrocki $q$,$t$ counting Dyck paths with forced and forbidden touch points I will give a combinatorial formula for certain coefficients of the operator $\nabla$ when it acts on a Hall-Littlewood symmetric functions. This result (almost completely) answers a conjecture posed by Alain Lascoux in the paper by (F)Bergeron-Garsia-Haiman-Tesler that introduced the operator $\nabla$. The combinatorial formula is proven by showing that $q$,$t$-counting Dyck paths satisfy the same recursion as a symmetric function expression. This is joint work with N. Bergeron, F. Descouens, A. Garsia, J. Haglund, A. Hicks, J. Morse and G. Xin.15 Feb. 2010 No seminar Family Day 8 Feb. 2010 Chris Berg Crystal models of the basic representation of the affine special linear Lie algebra Misra and Miwa gave a realization of the crystal of the basic representation in which the nodes of the graph are indexed by "$p$-regular" partitions. These "$p$-regular" partitions also index irreducible representations of the symmetric group over a field of characteristic $p$. After reviewing this, I will talk about some other partition based models for this crystal and their connections to the modular representation theory of the symmetric group.

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