This seminar has been running since 2002.
This seminar is partially supported by the grant Focus Research Group NSF 06580
The topics of talks have typically been any mixture of algebra with any other fields:
combinatorics, geometry, topology, physics, etc.
We usually have half a dozen to a dozen people in attendance at the seminars,
and this usually includes several graduate students and postdocs.
For this reason, we encourage the speakers to devote a portion of their talk to the suggestion of open problems
and the direction of research in their area.
This year, we will be interested (among other topics) in
generalizations of the LittlewoodRichardson rule.
 We will have a series of discussions aimed at understanding the PoirierReutenauer Hopf algebra of tableaux,
and its relations with symmetric functions.
In particular we will observe posets isomorphism arising from the product of Young tableaux.
 An interesting problem suggested by this posets isomorphism
is the one of finding an efficient algorithm to generate the posets,
this would also be an efficient algorithm for the multiplication of two Schur functions.
 Another question we will explore is related to a conjecture by
Fomin, Fulton, Li and Poon.
This conjecture states the Schurpositivity of a difference of two products of Schur functions.
We believe it would be appropriate to compare the corresponding posets,
and eventually prove their inclusion,
which will be enough to prove the conjecture.
 In another series of discussions, we will focus on shifted tableaux and QSchur functions.
 We will see how it is possible to realize an algebra of shifted tableaux,
which has essentially the same properties as the PoirierReutenauer Hopf algebra of Young tableaux.
In particuler, the LittlewoodRichardson rule for QSchur functions
may be combinatorially derived from the product of this algebra of shifted tableaux.
 We will also discuss and explore properties of the QHallLittlewood basis,
and the QKostka polynomials.




Archives 20082009

Amel Kaouche, University du Québec à Montréal (UQAM)
Please note that this talk will take place at Fields Institute !


 Title: Imperfect gases and graph invariants
 Date: Monday, November 24, 2008
 Time: 3:00p.m.
 Place: Fields Institute, Room 210
The Mayer and ReeHoover theories for the virial expansions in the context of a nonideal gas
reveal certain invariants (weights) associated to graphs.
We give a special attention to the case of the hardcore continuum gas in one dimension.
We present the method of graph homomorphisms that we apply to compute the Mayer and ReeHoover weights of various classes of graphs.




Luis Guillermo Serrano Herrera, University of Michigan 

 Title: The shifted plactic monoid
 Date: Friday, November 21, 2008
 Time: 3:30p.m.
 Place: Fields Institute, Room 210
We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger,
the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations,
or using Haiman's mixed insertion.
Applications include: a new combinatorial derivation (and a new version of)
the shifted LittlewoodRichardson Rule;
similar results for the coefficients in the Schur expansion of a Schur Pfunction;
and a shifted counterpart of the theory of noncommutative Schur functions in plactic variables.
A preprint is available at http://arXiv.org/abs/0811.2057.




Joel Kamnitzer, University of Toronto


 Title: MV polytopes and components of quiver varieties
 Date: Monday, November 17, 2008
 Time: 3:00p.m.
 Place: Fields Institute, Room 210
A number of interesting bases exist for the upper half of the universal envelopping algebra of a semisimple Lie algebra.
One such basis is Lusztig's semicanonical basis which is indexed by components of quiver varieties.
Another interesting basis is indexed by MirkovicVilonen cycles which lead to the combinatorics of MV polytopes.
In this talk, I will explain a natural bijection between the components of quiver varieties and the MV polytopes.
This is joint work with Pierre Baumann.


Janvier Nzeutchap, York University and Fields Institute


 Title: RobinsonSchensted Algorithm for Shifted Tableaux, PSchur and QSchur functions (part 2)
 Date: Friday, November 14, 2008
 Time: 3:30p.m.
 Place: Fields Institute, Room 210


Hugh Thomas, University of New Brunswick 

 Title: Antichains in the poset of positive roots, Catalan phenomena, and some conjectures of Panyushev
 Date: Monday, November 10, 2008
 Time: 3:00p.m.
 Fields Institute, Room 210


Associated to any finite crystallographic root system, there is a certain number,
the generalized Catalan number.
There are two major families of objects counted by the generalized Catalan number: one family contains the
clusters in the associated cluster algebra and the noncrossing partitions
in the associated reflection group, and others, while the second family
contains the antichains in the poset of positive roots, regions in the Shi
arrangement inside the dominant chamber, and others. There are bijections
within each family, but no natural typefree bijection between the
families. I will report on work towards constructing such a bijection.
It turns out that a crucial ingredient is a certain cyclic action on the
antichains in the poset of positive roots defined by Panyushev
(arXiv:0711.3353). This action is nontrivial to analyze even in type A.
In the course of our construction, we prove some of Panyushev's
conjectures about his action.
This is joint work with Drew Armstrong.


Janvier Nzeutchap, York University and Fields Institute 

 Title: RobinsonSchensted Algorithm for Shifted Tableaux, PSchur and QSchur functions
 Date: Friday, October 31, 2008
 Time: 3:30p.m.
 Fields Institute, Room 210


 analogue of RobinsonSchensted algoritm for shifted tableaux
 relation with the classical RobinsonSchensted algoritm, shifted Knuth equivalence
 combinatorial definition of PSchur and QSchur functions
 LittlewoodRichardson rule for PSchur and QSchur functions
 shifted analogue of the algebra of tableaux



Janvier Nzeutchap, York University and Fields Institute 

 Title: The PoirierReutenauer Hopf algebra of tableaux (part 2)
 Date: Friday, October 24, 2008
 Time: 3:30p.m.
 Fields Institute, Room 210


 review the definition of free Schur functions
 product of tableaux and concatenation of languages (coplactic classes)
 products of tableaux and posets isomorphism: a suspected bijection
 a Yamanouchi poset



Janvier Nzeutchap, York University and Fields Institute 

 Title: The PoirierReutenauer Hopf algebra of tableaux
 Date: Friday, October 17, 2008
 Time: 3:30p.m.
 Fields Institute library
 slides (pdf)


 review some essential definitions related to this algebra
 application 1: the LittlewoodRichardson rule
 introduce the permurohedron and tableauhedron orders
 application 2: tableaux posets and Kostka numbers
 recall a result due to Taskin: each product of two tableaux is an interval of the tableauhedron order
 state problem 2: an efficient algorithm to compute product of two Schur functions
 research direction 1 (introduction): partial order on Yamanouchi tableaux



Other Archives
20072008  
Open problems around kSchur functions and noncommutative symmetric functions 
20062007  
Open problems 
Fall 2005  Winter 2006  
Cluster Algebras and Quivers 
Spring 2005  
Formal languages and analytic classes of functions 
Fall 2004  
(Quasi) Symmetric functions in noncommutative variables and applications 
Winter 2003  
Crystal Bases and Representation Theory, Superalgebras, etc. 
Fall 2003  
QuasiSymmetric functions and applications 
Fall 2002  
Crystal Bases and Representation Theory 
 


