The Algebraic Combinatorics Seminar

Scheduled for Fridays at 3:30 PM, Room 210

A working seminar at the Fields Institute, director:
Nantel Bergeron

Date Speaker Title (click titles for abstract) Sept 15 Nantel Bergeron Organization, two problems:

1- symmetric functions in super space

2- The space of planar trees graded by signature.Sept 22 Nantel Bergeron and Rafael S. González D'león The space of trees graded by signature. Sept 29 Nantel Bergeron and Rafael S. González D'león The space of trees graded by signature. Oct 6 Mike Zabrocki Symmetric functions in super space I organized the big picture related to the goal of this project: there are a number of polynomial and non-commutative polynomial rings for which the invariants and quasi-invariants are well known Hopf algebras (Sym, QSym, NCSym, etc.). I described the minimal information that we need in order to implement Sym and QSym and left the symmetric functions in super space and the quasi-symmetric functions in super-space as the next steps.Oct 13

At YORKJerzy Wejman This will be a coloquium talk at YORK University. This week only Oct 20 Mike Zabrocki Symmetric functions in super space The following papers will be our main focus:Also my code is can be accessed by executing the command:

- Classical symmetric functions in superspace
- Pieri rules for Schur functions in superspace
- Jack superpolynomials, superpartition ordering and determinantal formulas
- Jack superpolynomials: physical and combinatorial definitions
- Orthogonality of Jack polynomials in superspace
- Macdonald polynomials in superspace as eigenfunctions of commuting operators
- N>=2 symmetric superpolynomials

load("http://garsia.math.yorku.ca/~zabrocki/sf_superspace.py")and the code for superpartitions is at:http://garsia.math.yorku.ca/~zabrocki/sf_superspace.pyandhttp://garsia.math.yorku.ca/~zabrocki/superpartitions.py

Oct 27 Reading week Nov 3 Rafael D'leon and Nantel Bergeron Hopf algebra of planar trees and stirling permutations (open questions in abstract) Last time we define Hopf algebra on Planar Binary Trees using a process describe by Aguiar and Sotille using "lightening". This is isomorphic to Loday-Ronco Hopf algebra. We can label the binary trees incrisingly and the "lightening" process still work to define a Hopf algebra. This time it is the larger and correspond to permutations. It is the Malvenuto-Reutenauer Hopf algebra. It is nice to express everything in term of trees as it makes the operation more transparent and explain better the relation between those algebras. Now we can generalize as well very easily. Consider non-degenerate planar trees (no internal node of degree 1). The process of lightening still work well and we can define two Hopf algebras on those trees We can also consider increasing labeling of such trees and now many natural questions arises: SEE QUESTIONS transcripted by JamesNov 10 Mike Zabrocki Symmetric functions in super space Last time we conjectured a formula for the Monomial and Fundamental quasi-symmetric functions in super space. We will need to work to refine and clarify the notation for these formulae and conjecture their product and coproduct in analogy with the monomial and fundamental bases of QSym.

Code for SuperCompositions and QuasiSymmetricFunctionsinSuperSpace (Monomial basis only) in Sage:

load("http://garsia.math.yorku.ca/~zabrocki/supercompositions.py")

load("http://garsia.math.yorku.ca/~zabrocki/superpartitions.py")

load("http://garsia.math.yorku.ca/~zabrocki/qsf_superspace.py")

load("http://garsia.math.yorku.ca/~zabrocki/sf_superspace.py")

Nov 17 Rafael D'leon and Nantel Bergeron Hopf algebra of planar trees, stirling permutations and parking functions. Nov 24 James Huang and Laura Colmenarjo P partition and Schur function in Symmetric functions in super space Dec 1 Marcelo Aguiar

[Cornell U.]Integral Hyperplane Arrangements Dec 8 Rafael D'leon and Nantel Bergeron Hopf algebra of planar trees, stirling permutations and parking functions.

Date Speaker Title (click titles for abstract) Jan 19 Laura Colmenarejo Quasisymmetric functions in superspace. We review the work of Luc Lapointe and his colaborators on quasisymmetric functions in super space.Jan 26 NO SEMINAR We attend CAAC conference in McMaster Feb 2 Nantel Bergeron A new working topic (see link to documents bellow) This winter we will continue to work on

(a) quasysymmetric functions in superspace

(b) Hopf algebra of trees (when Rafael is available)

(c) A new topic... We will find out on This day! I am now attaching document and more to come

Picture of a permutahedron.

This is a PDF that contains the question

This is a PDF that contains picture and some partial solutionFeb 9 Mike Zabrocki Harmonics in super space We will look at the quotient of the polynomial ring in a commuting and anti-commuting set of variables mod the ideal generated by the symmetric functions in superspace.Feb 16 Mike Zabrocki Harmonics in super space We look at the Frobenius series of the harmonics in super space.Feb 23 NO SEMINAR (reading week) Mar 2 Nantel Bergeron Kernel of $\Theta\circ\Psi\circ\Psi^*\circ\Theta^*$. We define the Hopf monoid involved and some of the maps.Mar 9 Mike Zabrocki Harmonics in super space. Continuing with the super-harmonic space we will talk about conjectures and techniques for finding formulas for the Hilbert series and Frobenius series. (data)Mar 16 Rafael Gonzales D'leon and Nantel Bergeron Continuing project of Hopf algebra of planar trees we start at 3:10... trying to connect with Rafael

- Unless otherwise indicated, the seminar meets on Fridays at 3:30 pm in Room 210 of the Fields Institute.
- If you are interested in speaking at the seminar, contact Nantel Bergeron
- You may also be interested in the Applied Algebra Seminar (Monday afternoons at York University).

** About the seminar. **
Every year we pick a new topics to explore.