| 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 fdescoue@fields.utoronto.ca |
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
| Date |
Speaker |
Subject |
| 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
Team
|
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 |
TBA |
TBA |
| 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 |
NO SEMINAR |
|
| 4 April 2008 |
NO SEMINAR |
|
| 11 April 2008 Special Session at 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. END OF THE SEMINAR FOR THIS YEAR
....
|
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 |