SCIENTIFIC PROGRAMS AND ACTIVITIES

November 27, 2014

Algebraic Combinatorics Seminar 2008-09
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. More informations or a suggestion of talk ? Don't hesistate contact bergeron(@)mathstat.yorku.ca

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

Date

Speaker

SPECIAL LECTURE
January 26
3:00
Carolina Benedetti (U. of Comlombia, South America)
Volumes of matroid polytopes
Given a matroid M we can associate with it its matroid polytope $P_M$ as well as its independent set polytope I_M. In this talk I will show an explicit way to decompose these polytopes as a signed Minkowski sum of simplices. Using this decomposition, which involves a lot of information of M such as its beta invariant, I will give a nice formula to calculate the volumes of P_M and I_M, offering a geometric point of view for beta (M). Finally, we will see analogous results in the case of a nice class of flag matroids, namely the cascading flag matroids. In this case the role of beta (M) will be done by the gamma invariant. This is joint work with Federico Ardila (San Francisco State University) and Jeffrey Doker (UC Berkeley).
January 9, 2009 Alexander Rossi Miller, University of Minnesota
Differential posets and Smith normal forms

In this talk I will introduce r-differential posets, then move on to some conjectures about their structure. The main focus will be on a conjecture asserting a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. We will discuss the current progress of this conjecture in Young's lattice, as well as in Young-Fibonacci posets. This is joint work with Vic Reiner.

A preprint is available at http://arxiv.org/abs/0811.1983

SPECIAL LECTURE
November 24
3:00

Amel Kaouche, University du Québec à Montréal (UQAM)
Imperfect gases and graph invariants

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.

November 21
3:30

Luis Guillermo Serrano Herrera, University of Michigan
The shifted plactic monoid

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.

SPECIAL LECTURE
November 17
3:00 p.m.
Joel Kamnitzer, University of Toronto
MV polytopes and components of quiver varieties

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.
November 14
3:30
Janvier Nzeutchap, York University and Fields Institute
Robinson-Schensted Algorithm for Shifted Tableaux, P-Schur and Q-Schur functions (part 2)

SPECIAL LECTURE
November 10
3:00 p.m.

Hugh Thomas (University of New Brunswick)
Antichains in the poset of positive roots, Catalan phenomena, and some conjectures of Panyushev

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.
October 31
3:30
Janvier Nzeutchap, York University and the Fields Institute
Robinson-Schensted Algorithm for Shifted Tableaux
October 24
3:30

Janvier Nzeutchap, York University and the Fields Institute
The Poirier-Reutenauer Hopf algebra of tableaux (part 2)

During this session, we will use many elementary examples.
1. review the definition of plactic Schur functions
2. product of tableaux and concatenation of languages (coplactic classes)
3. products of tableaux and posets isomorphism: a suspected bijection
4. a Yamanouchi poset

October 17
3:30
Janvier Nzeutchap, York University and the Fields Institute
The Poirier-Reutenauer Hopf algebra of tableaux

During this session, we will use many elementary examples.
1. review some essential definitions related to this algebra
2. application 1: the Littlewood-Richardson rule
3. introduce the permurohedron, dominance and tableauhedron orders
4. application 2: tableaux posets and Kostka numbers
5. application 3: an interpretation of homogenous symmetric functions
6. recall a result due to Taskin: each product of two tableaux is an interval of each tableaux poset
7. state problem 1: describe cover relations in the dominance and tableauhedron orders
8. state problem 2: an efficient algorithm to compute product of two Schur functions
9. research direction 1 (introduction): products of tableaux and posets isomorphism