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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)
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PAST SEMINARS
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| Sept. 25, 2009 |
Mike Zabrocki (York University)
Decomposing GLn-modules into Sn-modules I
TBA
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| Oct. 2, 2009 |
Mike Zabrocki (York University)
Decomposing GLn-modules into Sn-modules II
TBA
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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].
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| January 22, 2010 |
No Seminar
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| 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.
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| 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.
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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.
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| 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.
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| 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.
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| 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.)
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