February 23, 2019

Algebraic Combinatorics Seminar 2007-08
held at the Fields Institute

Fridays 4 p.m.-5:30 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 to contact the organizing committee Francois Descouens (fdescoue AT or Nantel Bergeron (York).

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

6 June 4 p.m.
(Special Session)

Anouk Bergeron-Brlek

18 April 4 p.m.
(Special Session)
Muge Taskin (Fields)
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.
11 Apr. 4 p.m.
(Special Session)
Muriel Livernet (MIT)
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 ofConnes and Kreimer in renormalisation theory and operads built on rooted trees.
22 Feb. 4 p.m. 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.
15 Feb. 4 p.m. TBA
8 Feb. 4 p.m. Nantel Bergeron
New developements on the filtration of diagonal harmonics
1 Feb. 4 p.m. 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.-

25 Jan. 4 p.m. Working session focused on new developments about generalizations of q,t-Catalan numbers
11 Jan. 4p.m. 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.
30 Nov. 4p.m. York/ Fields Combinatorics Team
Open problems - III
23 Nov. 4p.m. York/ Fields Combinatorics Team
Open problems - II
16 Nov. 4p.m. York/ Fields Combinatorics Team
Open problems - I
9 Nov - 4p.m. Huilan Li, York University
Representation theory of the Hecke algebra at q=0
2 Nov - 4p.m. Lenny Tevlin (Yeshiva University, New York)
Noncommutative Cauchy and q-Cauchy Identities
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 seemsnatural 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.
26 Oct - 4p.m. Francois Descouens (Fields Institute)
Experimentations on Non-commutative symmetric functions with MuPAD
19 Oct- 4p.m.

Mike Zabrocki (York University)

12 Oct- 4p.m. Mike Zabrocki (York University)
Analogs of k-Schur functions in NCSF
5 Oct.- 4 p.m.

Mike Zabrocki (York University)

28 Sep- 3 p.m.** time change Nantel Bergeron (York University)
Introduction on Non-Commutative Symmetric Functions
21 Sep- 4 p.m. Nantel Bergeron (York University)
Introduction to Non Communative Symmetric Functions