Algebraic Combinatorics Seminar
on Friday afternoon at 4pm at the Fields Institute (2nd Floor, Room 210)

The Fields Institute for Research in Mathematical Sciences
222 College Street · Toronto, Ontario · M5T 3J1 · Canada
2007-2008 Topics : Open problems around k-Schur functions and non-commutative symmetric functions

More informations or a suggestion of talk ? Don't hesistate to contact Francois Descouens

This seminar is partially supported by the grant
Focus Research Group NSF 06-580:

« Affine Schubert Calculus: Combinatorial, geometric, physical and computational aspect ».

This many-faceted project involves and ties together various problems from combinatorics, geometry, representation theory, physics, and computation.  Ones of the main questions deals about k-Schur functions. The subject of this seminar is to study non-commutative analogs of these functions and more generaly we will study non-commutative symmetric functions and the representation theory of the 0-Hecke algebra.

We also organize special sessions in coordination with the Applied Algebra Seminar of York University

Programme of the year

The lectures notes of N. Bergeron and M. Zabrocki are here (Thanks to Huilan Li who carrefully takes them)
Some notes on the open problems session are here(Thanks again to Huilan)

21 Sept 2007
Nantel Bergeron
(Yok University)
Introduction to non-commutative symmetric functions (NCSF) - I
28 Sept 2007
Excep.  at 3 p.m

Nantel Bergeron
 (York University)
 Introduction to non-commutative symmetric functions (NCSF) - II
5 October 2007
Mike Zabrocki
 (York University)
Analogs of Hall-Littlewood and Macdonald functions in NCSF
12 October 2007 Mike Zabrocki
 (York University)
Analogs of k-Schur functions in NCSF
19 October 2007
Mike Zabrocki (York University)
Tutorial MuPAD
26 October 2007
    Francois Descouens
 (Fields Institute and York University)
Experimentations on Non-commutative symmetric functions with MuPAD
2 Nov 2007
Special Session

Lenny Tevlin
(Yeshiva University, New York)
Noncommutative Cauchy and q-Cauchy Identities.

Abstarct: In the talk I will try draw a parallel between the classical theory of symmetric functions and that of
noncommutative ones.  In particular there are two new bases in NSym, the analog of monomial and fundamental
bases, that allow one to introduce an analog of the classical Cauchy identity. It appears that in the noncommutative
world both ribbon Schur and fundamental functions are distinct analogs of classical Schur functions. Integrality of
ribbon Schur basis in either of monomial or fundamental noncommutative basis  (which has been recently proven)
requires an introduction of what appears to be an interesting new statistics on permutations. Therefore it seems
natural to expect new interesting objects to arise with q- and q,t generalizations. However, from the point of view
of the present writer the situation with noncommutative Hall-Littlewood functions is less complete and I will discuss
existing candidates apropos a noncommutative q-Cauchy identity.
9 Nov 2007
Huilan Li
(York University)
Representation theory of the Hecke algebra at q=0
16 Nov 2007
York/ Fields Combinatorics Team
Open problems - I
23 Nov 2007
York/ Fields Combinatorics Team
Open problems - II
30 Nov 2007
York/ Fields Combinatorics Team
Open problems - III
11 Jan 2008
 Special Session
Nick Loehr
(Virginia Tech)
Combinatorial Aspects of the Bergeron-Garsia Nabla Operator

The nabla operator introduced by Francois Bergeron and Adriano Garsia plays a key role in the theory of symmetric
functions and Macdonald polynomials. Over the past decade, many advances have been made in our understanding
of the combinatorial significance of the nabla operator. This talk will survey recent research in this area, beginning with
the "q,t-Catalan Theorem" of Garsia, Haglund, and Haiman and ending with a new conjectured formula for the image
of any Schur function under nabla (which is joint work with Greg Warrington). Along the way, we will encounter many
fascinating combinatorial and algebraic entities, including parking functions, quantum lattice paths, LLT polynomials,
diagonal harmonics modules, and Macdonald polynomials.
18 Jan 2008

NO SEMINAR (canadian workshop in Halifax)
25 Jan 2008
York/Fields Combinatorics
Working session focused on new developments about generalizations of q,t-Catalan numbers
1 Feb 2008
Special Session
Mahir Can
(University of Western Ontario)
Some plethystic identities regarding the diagonal harmonics module.

The Garsia-Haglund proof of the (q,t)-Catalan conjecture makes use of plethystically defined, still mysterious, symmetric
functions $E_{n,k}$. In this talk, we present several symmetric function identities involving the functions $E_{n,k}$.
If the time permits, We will also talk (speculate) about a seemingly forgotten conjecture  of Garsia and Haglund on the
sectionalization of the diagonal harmonics module.

-No prior background on the subject is expected.-
8 Feb 2008
Nantel Bergeron
New developements on the filtration of diagonal harmonics
 15 Feb 2008
22 Feb 2008
Special Session
John Irvine
(Saint Mary's University, Halifax)
 Counting Lattice Paths Under a Shifting Boundary

The generalized ballot theorem gives a well-known formula for the number of lattice paths in the first quadrant lying weakly
under the line x=ay, where a is an arbitrary positive integer.  While there is no simple formula for the number of paths under
an arbitrary piecewise linear boundary, we show that nice enumerative results are available if we allow for cyclic shifts of such
 a general boundaries. We show how our formula quickly yields recent results concerning paths dominated  by periodic boundaries. 
A refinement allows for the counting of paths with a specified number of corners.  This is joint work with A. Rattan. 
29 Feb 2008
Francois Descouens
(Fields Institute)
Combinatorial interpretation for generalized q,t-Catalan numbers of level 2 and n-3
7 March 2008
York / Fields Combinatorics Team
Open problems on q,t-Catalan numbers and their generalizations - I
14 March 2008
York / Fields Combinatorics Team Open problems on q,t-Catalan numbers and their generalizations - II
  21 March 2008

NO SEMINAR (Good Friday)
28 March 2008

4 April 2008

11 April 2008
Special Session

4.30 pm
Muriel Livernet
(Paris XIII)
Posets, Groups and Hopf algebras associated to a set-operad.

In this talk we will review a  result of Bruno Vallette linking the notion of Koszul duality for operads and Cohen-MacCauley posets.
We'll present in this context a joint work with F. Chapoton, where we compare two Hopf algebras, one built directly from operads,
and another one associated to the incidence Hopf algebra of a family of posets. This leads us to a new link between the Hopf algebra of
Connes and Kreimer in renormalisation theory and operads built on rooted trees.

18 April 2008
Special Session
Muge Taskin
(Fields Institute)
Plactic relations for r-domino tableaux

The recent work of Bonnafé et al. (2007) shows through two conjectures that r-domino tableaux have an important role in
Kazhdan-Lusztig theory of type B with unequal parameters. In this paper we provide plactic relations on signed permutations
which determine whether given two signed permutations have the same insertion r-domino tableaux in Grafinkle’s algorithm (1990).
Moreover, we show that a particular  extension of these relations can describe Garfinkle’s equivalence relation on r-domino tableaux
which is given through the notion of open cycles.  With these results we articulate the conjectures of Bonnafé, Geck, Iancu, and Lam by providing necessary tools for their proof.


Past Topics

Fall 2006
Open problems
Fall 2005
Reading sessions on cluster algebras
Spring 2005
Formal languages and analytic classes of functions
Fall 2004
(Quasi-) Symmetric functions in noncommutative variables and applications
Winter 2003
Topic Crystal bases for superalgebras
Fall 2003 Quasi-symmetric functions and applications
Fall 2002
Topic Crystal bases