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This seminar has been running since 2002.
This seminar is partially supported by the grant Focus Research Group NSF 06-580
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 post-docs.
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 Littlewood-Richardson rule.
- We will have a series of discussions aimed at understanding the Poirier-Reutenauer 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 Schur-positivity 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 Q-Schur functions.
- We will see how it is possible to realize an algebra of shifted tableaux,
which has essentially the same properties as the Poirier-Reutenauer Hopf algebra of Young tableaux.
In particuler, the Littlewood-Richardson rule for Q-Schur functions
may be combinatorially derived from the product of this algebra of shifted tableaux.
- We will also discuss and explore properties of the Q-Hall-Littlewood basis,
and the Q-Kostka polynomials.
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Archives 2008-2009
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Amel Kaouche, University du Québec à Montréal (UQAM)
Please note that this talk will take place at Fields Institute !
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- Title: Imperfect gases and graph invariants
- Date: Monday, November 24, 2008
- Time: 3:00p.m.
- Place: Fields Institute, Room 210
The Mayer and Ree-Hoover theories for the virial expansions in the context of a non-ideal gas
reveal certain invariants (weights) associated to graphs.
We give a special attention to the case of the hard-core continuum gas in one dimension.
We present the method of graph homomorphisms that we apply to compute the Mayer and Ree-Hoover weights of various classes of graphs.
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Luis Guillermo Serrano Herrera, University of Michigan |
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- 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 Littlewood-Richardson Rule;
similar results for the coefficients in the Schur expansion of a Schur P-function;
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.
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Joel Kamnitzer, University of Toronto
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- 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 Mirkovic-Vilonen 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.
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Janvier Nzeutchap, York University and Fields Institute
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- Title: Robinson-Schensted Algorithm for Shifted Tableaux, P-Schur and Q-Schur functions (part 2)
- Date: Friday, November 14, 2008
- Time: 3:30p.m.
- Place: Fields Institute, Room 210
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Hugh Thomas, University of New Brunswick |
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- 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
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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 type-free 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 non-trivial 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.
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Janvier Nzeutchap, York University and Fields Institute |
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- Title: Robinson-Schensted Algorithm for Shifted Tableaux, P-Schur and Q-Schur functions
- Date: Friday, October 31, 2008
- Time: 3:30p.m.
- Fields Institute, Room 210
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- analogue of Robinson-Schensted algoritm for shifted tableaux
- relation with the classical Robinson-Schensted algoritm, shifted Knuth equivalence
- combinatorial definition of P-Schur and Q-Schur functions
- Littlewood-Richardson rule for P-Schur and Q-Schur functions
- shifted analogue of the algebra of tableaux
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Janvier Nzeutchap, York University and Fields Institute |
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- Title: The Poirier-Reutenauer Hopf algebra of tableaux (part 2)
- Date: Friday, October 24, 2008
- Time: 3:30p.m.
- Fields Institute, Room 210
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- 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
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Janvier Nzeutchap, York University and Fields Institute |
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- Title: The Poirier-Reutenauer Hopf algebra of tableaux
- Date: Friday, October 17, 2008
- Time: 3:30p.m.
- Fields Institute library
- slides (pdf)
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- review some essential definitions related to this algebra
- application 1: the Littlewood-Richardson 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
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Other Archives
| 2007-2008 | |
Open problems around k-Schur functions and non-commutative symmetric functions |
| 2006-2007 | |
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, Super-algebras, etc. |
| Fall 2003 | |
Quasi-Symmetric functions and applications |
| Fall 2002 | |
Crystal Bases and Representation Theory |
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