The Applied Algebra Seminar

Schedule

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
TBA

05 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
TBA

22 Mar. 2010
TBA

15 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 define drift configurations and 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.

Fall 2008 & Winter 2009	Fall 2007	Winter 2006	Fall 2005
Winter 2005	Fall 2004	Winter 2004	Fall 2003
Winter 2003	Fall 2002	Winter 2002	Fall 2001
Winter 2001	Fall 2000	Winter 2000	Fall 1999
Winter 1999	Fall 1998	Winter 1998	Fall 1997

About the seminar

Schedule

Archives

Date	Speaker	Title (click titles for abstract)
12 Apr. 2010	Ada Chan	Type II matrices TBA
05 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 TBA
22 Mar. 2010		TBA
15 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 define drift configurations and 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.