The Algebraic Combinatorics Seminar
Scheduled for Fridays at 3:30 PM, Library or Room 210
A working seminar at the Fields Institute organized by Nantel Bergeron , Robin Sulzgruber and John Machacek.

#### Schedule FALL 2019 Click to see/hide schedule

Date Speaker Title (click titles for abstract)
Sept 6 R. Sulzgruber (York) First meeting Please come and present a problem you would like to work on this year
Each presenter will be given 15 minutes to introduce a problem.

Mike: Super space quotient is ${\mathbb Q}[x_1,...,x_n,\theta_1,...,\theta_n]$ modulo the ideal of diagonal symmetric function. The x commutes, the $\theta$ anticommute.
Conjecture: dim=number of set composition of $\{1,2,...,n\}$
Conjecture: Frobenius characteristic is $\sum_{k=0}^{n-1} z^{n-k+1} \Delta'_{e_k}(e_n)\Big|_{q=0}$
Problem: give a description of the Grobner basis of the ideal

John: Given a 2-simplex the face poset P is the truncated (no empty face) boolean lattice of 3 elements. Let $\Delta(P)$ be the (chain) simplicial complex of $P$ (6 triangle inside a triangle)
the f vector is f=(1,7,12,6) h=(1,4,1,0) (the eulerian number)
Now look at $RP^2$, do the sign decomposition of the space [remark in projective space (-,0,-)=(+,0,+) so the sign cells are indexed by sequences of +,0,- where the first non-zero in +] Let P be the poset of adjacency (removing (0,0,0)) and compute h-vectoir of $\Delta(P)$
Conjecture: The h-vector is a type D eulerian number (looking at Dn inside Bn)

Shu Xiao: $PF_n$ are function X:[n]->[n] such that the number of $X(i)<=j$ is more than j
$PF_2= \{11,12,21\}$
Given two parking functions X and Y of n and m we can do the shifted concatenation XY and get a parking function of n+m. A parking function is connected if it is not obtain by a shifted concatenation of two smaller ones
Now let $PF'_n$ be the set of binary trees with n+1 leaves labeled by a permutation $\sigma$ such that right-leaves are descent of $\sigma$
A global descent of a pair $(t,\sigma)$ if we can cut the tree t on the leftmost branch, and it is the global descent of the permutation labeling.
Problem: Find a bijection between the connected parking function and the set of labelled tree with no global descent.

Kelvin: Problem. Define $F_\pi (X,q) =\sum_{(w,\pi) word parking functions} q^{dinv(w,\pi)} X^{w}$.
Problem: Prove bijectively that this is in $Sym\otimes Q(q)$
$(w,\pi)$ is a valid word parking function if the word w labelled the up step of $\pi$ a catalan path and the labelled are strictly increasing in column. The dinv is defined in picture.

Mike: Diagram algebras. One example, the partition algebra is an algebra of graphs on [2k] (draw two rows of n elements and draw components of a set partition of [2k], we can multiply by juxtaposition of to such drawing.)
What is the representation theory of such algebra? It is indexed by integer partitions of $\le k$ known Branching rule describe how is the representation of $P_k$ relates to $P_{k-1}$. This rules tells us that the dimension of the irreducible is the number of paths in the branching rule graph from the bottom to the given irreducible. This can be indexed by some set valued tableaux of shape lambda and content {1,2,...,k}
The Tenabe algebra inside P_k (given by some condition on the size of the blocks)
Question: What is the subset of set valued tableaux that give the dimension of the irreducible of this algebra.
Sept 13 R Sulzgruber (York) Parking functions.
Sept 20
Sept 27
Oct 4
Oct 11
Oct 18 Reading Week No Seminar
Oct 25
Nov 1
Nov 8 SPECIAL AAS
Sami Assaf (USC)
A Pieri rule for key polynomials
Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for GL_n. The Pieri rule for the product of a Schur function and a single row Schur function is a multiplicity-free branching rule with a beautiful combinatorial interpretation in terms of adding boxes to a Young diagram. Key polynomials are an interesting basis of the polynomial ring originally defined as characters of submodules for irreducible GL_n modules under the action of upper triangular matrices. In this talk, I'll present joint work with Danjoseph Quijada where we give a Pieri rule for the product of a key polynomial and a single row key polynomial. While this formula has signs, it is multiplicity-free and has an interpretation in terms of adding balls to a key diagram.
Nov 15
Nov 22
Nov 29
dec 6

#### Schedule WINTER 2020 Click to see/hide schedule

Date Speaker Title (click titles for abstract)
Jan 10
Jan 17
Jan 24
Jan 31
Feb 7
Feb 14
Feb 21 NO SEMINAR Reading Week
Feb 28
Mar 6
Mar 14
Mar 20
Mar 27

Notes.

• 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.

Year Topic
2018-2019 Super Harmonics, steep bounce and a little bit of q-fibonomials.
2017-2018 (Quasi)symmetric functions in superspace, Hopf algebra of planar trees
2016-2017 Quantum Schubert; The theta map; Reduce order of symmetric group; Branching rule between GL(n) and symmetric group.
2015-2016 Positroid and Matroid/ Hopf algebras and quotients.
2014-2015 Fiboland, Symmetric and non symmetric functions
2013-2014 Fiboland, a world of Catalan and Fibonacci numbers
2012-2013 NSym and the Immaculate Basis
2011-2012 k-Schur functions and affine permutations
2010-2011 Littlewood Richardson rule k-Schur functions.
2009-2010 Idempotents and weakly ordered semigroups. (q,t) Catalan Numbers.
2008-2009 Littlewood-Richardson Rule, Shifted Tableaux and P-Schur functions
2007-2008 Open problems around k-Schur functions and non-commutative symmetric functions
2006-2007 Open problems
2005-2006 Cluster Algebras and Quivers
Spring 05 Formal languages and analytic classes of functions
Fall 04 (Quasi-) Symmetric functions in noncommutative variables and applications
Winter 03 Crystal Bases and Representation Theory, Super-algebras, etc.
Fall 03 Quasi-Symmetric functions and applications
Fall 02 Crystal Bases and Representation Theory