SCIENTIFIC PROGRAMS AND ACTIVITIES

September 18, 2014

Algebraic Combinatorics Seminar 2009-10
held at the Fields Institute

Fridays 3:30 p.m.-5:00 p.m., unless otherwise indicated.

The purpose of this seminar is to cover exposition on topics of algebraic combinatorics which are of interest to the people attending, so please feel free to come and participate. Every year we pick a new topic to explore. We will be selecting the seminar topic for this year shortly, so attend the first few talks if you want to influence the decision.
If you are interested in speaking at the seminar, contact Franco Saliola (saliola@gmail.com).


We also organize special sessions jointly with the Applied Algebra Seminar (York University)

PAST SEMINARS
Sept. 25, 2009

Mike Zabrocki (York University)
Decomposing GLn-modules into Sn-modules I

TBA

Oct. 2, 2009

Mike Zabrocki (York University)
Decomposing GLn-modules into Sn-modules II

TBA

Oct. 9, 2009 Franco Saliola
Random-to-random shuffles and commuting families of matrices

We investigate a family of matrices with rows and columns indexed by permutations and with entries that count the number of increasing subsequences appearing in the permutations. These matrices are related to the inversion statistic on permutations, the Varchenko matrix for the reflection arrangement of the symmetric group, the linear-ordering polytope, and the transition matrix for the random-to-random shuffle. We will explore the connections with random walks on hyperplane arrangements and with the representation theory of the symmetric group, explaining how these can be used to study the eigenvalues and eigenspaces of the matrices.

This is joint work with Victor Reiner and Volkmar Welker.

Oct. 16, 2009 No Seminar
Oct. 23, 2009 Franco Saliola
Random-to-random shuffles and commuting families of matrices II

We will continue our investigation of families of matrices with rows and columns indexed by permutations and with entries given by a certain statistics on permutations. For those who missed the first talk, this talk will begin by quickly recalling the necessary definitions and tools. We will then proceed to investigate properties of the matrices (commutation, eigenvalues, eigenspaces, ...).
This is joint work with Victor Reiner and Volkmar Welker.

Oct. 30, 2009 Open Problem Session
Nov. 6, 2009 Working Seminar
Nov. 13, 2009

Mike Zabrocki
Decomposing GLn-modules into Sn-modules III: Progress

I will present some recent progress on the problem of decomposing GLn-modules into Sn-modules.

Nov. 16, 2009 **SPECIAL SEMINAR**

Igor Pak, UCLA
Random standard Young tableaux

I will discuss the problem of generating random standard Young tableaux of a given shape. I will then define and analyze a weighted hook walk, which is a multivariable deformation of the usual hook walk. Finally, I will show how this weighted deformation gives a new bijective proof of the hook length formula for the number of standard Young tableaux. This is joint work with Ionut Ciocan-Fontanine and Matjaz Konvalinka.
Nov. 20, 2009 No Seminar
Nov. 27, 2009

Franco Saliola
Random Walks on Hyperplane Arrangements

I will present a new derivation of the main results of the Bidigare-Hanlon-Rockmore theory of random walks on hyperplane arrangements. The approach is to use an idea from the work of Ken Brown: consider the probability distribution as an element in a semigroup algebra and use algebraic techniques to study the random walk. I will introduce a recursive construction of orthogonal idempotents and explain how this construction produces orthogonal idempotents decomposing the probability element.

Dec. 4, 2009 No Seminar
Dec. 11, 2009

Anouk Bergeron-Brlek
Words on non-commutative invariants of the hyperoctahedral group

Consider the hyperoctahedral group B_n and let V be a vector space which has a B_n-module structure. We present a general combinatorial method to decompose the tensor algebra T(V) on V into irreducible modules in terms of words in a particular Cayley graph of B_n. We make explicit the example of V being the geometric module (corresponding to the action of B_n as a reflection group) and give combinatorial interpretations for the graded dimensions and the number of free generators of the subalgebra T(V)^{B_n} of invariants of B_n, in term of those words.

January 15, 2010

Nantel Bergeron (York University)
Radical of Weakly Ordered Semigroup Algebras
We define the notion of weakly ordered semigroups. For this class of semigroups, we compute the radical of the semigroup algebras. This generalizes some results on left regular bands and on 0- Hecke algebras. One open problem is to give a construction of the minimal idempotent for the 0-Hecke algebra [Our hope is to use generalize (for Weakly Ordered Semigroup Algebras) the technique presented by Franco for left regular bands].

January 22, 2010

No Seminar

January 29, 2010 Chris Berg
(qt) -Catalan numbers and Cores
Anderson gave a bijection between Dyck paths and n-cores which are also (n+1)-cores. I will describe this bijection and explain how to calculate the dinv and area statistics on these cores. More recently, Vazirani and Fishel gave a bijection between n and nm+1 cores and shi arrangements. I will define statistics on these cores in an attempt to give a formula for the Catalan Fuss polynomials.
Feb. 5, 2010 Chris Berg and Mike Zabrocki
Exploring (qt) -Catalan numbers and Cores
We will continue to explore the bijections from the previous seminar. The aim is to understand various operations on cores and different ways to compute the statistics in the new settings.
Feb. 12, 2010 Ton Dieker, School of Industrial and Systems Engineering, Georgia Tech
Interlacings, representation theory, and the interchange process on weighted graphs
A central question in the theory of card shuffling is how quickly a deck of cards becomes 'well-shuffled' given a shuffling rule. Using basic tools from the representation theory of the symmetric group, I will discuss a probabilistic card shuffling model known as the 'interchange process'. A 1992 conjecture by Aldous and Diaconis about this model has recently been resolved (see http://www.stat.berkeley.edu/~aldous/Research/OP/sgap.html) and I will indicate how my work has been involved with this.
Feb. 17, 2010
3:15 p.m.
Stewart Library
*Please note special date*
Benjamin Steinberg (Carleton)
The representation theory of finite semigroups
In recent years the representation theory of certain classes of finite semigroups have been used successfully to study hyperplane chamber random walks and other Markov chains. In this talk we give a survey of semigroup representation theory intended for people working in algebraic combinatorics. We focus on the following aspects:

* Construction of the irreducible representations
* The character table
* The radical and triangularizability

If time permits we will discuss some applications to probability and automata theory.

Feb. 26, 2010 Working Seminar
Mar. 12, 2010 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.

Mar. 19, 2010

A discussion on k-Schur functions and cores

The upcoming Algebraic Combinatorics Seminar will feature a discussion on the relationship between k-Schur functions and cores. It will begin with a presentation of the definition of k-Schur functions and cores, for those who are not familiar with these objects. The discussion will be lead by Mike Zabrocki.

Mar. 26, 2010

Sonya Berg (UC Davis)
A quantum algorithm for the quantum Schur transform

The quantum Schur transform is a unitary implementation of q-deformed Schur-Weyl duality in type A. I'll present an efficient quantum algorithm for its computation, and explain its relationship to RSK algorithms. (Note the double use of the word quantum: one for q-deformed algebras and one for quantum computation.)