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.

Date Speaker Title (click titles for abstract) Sept 6 Kelvin Chan, Shu Xiao Li, John Machacek, and Mike Zabrocki (all 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 Robin Sulzgruber (York) Parking functions I will give an overview of the different combinatorial realisations of Dyck paths and parking functions and of some directions in which they can be generalised. Then I will construct a bijection between the two types of connected parking functions that were introduced by Shu Xiao last week.Sept 20 John Machacek (York) Sign variation and h-vectors We will revisit the problem of finding the h-vector of certain triangulations inside real projective space. Notions from combinatorial topology and Coxeter groups will be reviewed.Sept 27 Kelvin Chan (York) Symmetry of LLT Polynomials In the theory of Macdonald polynomials, the LLT family lives on a busy street. Relevant facts include "expansions" of modified Macdonld polynomials and $\nabla e_{n}$. Interestingly, one inevitably questions its symmetry upon first contact. We will discuss some of their properties and an induction proof of their symmetry by Haglund, Haiman and Loehr.Oct 4 John Campbell (York) Schur-Weyl duality for diagram algebras, and a problem involving Tanabe algebras Schur-Weyl duality refers to the duality between the canonical actions of $\text{GL}_n(\mathbb{C})$ and $S_k$ on tensor spaces of the form $V^{\otimes k}$, for a $n$-dimensional vector space $V$. If we consider the symmetric group $S_n$ as a subgroup of $\text{GL}_n(\mathbb{C})$, this gives us an instance of Schur-Weyl duality, in this case between $S_n$ and an algebra $\mathbb{C}A_k(n)$ known as the partition algebra, which is isomorphic to $\text{End}_{S_n}(V^{\otimes k})$ for $2k \leq n$. Partition algebras may be defined using an operation known as diagram multiplication, and this talk will cover how some of the basic properties of partition algebras may be derived using Schur-Weyl duality. The problem of applying some of the main concepts from the preprint arxiv.org/abs/1905.02071 to Tanabe algebras will also be considered.Oct 11 Mike Zabrocki (York) Centralizer algebras, Bratteli diagrams and set valued tableaux I'll give some examples of pairs of centralizer algebras acting on a space and explain how the representation theory of the algebras relate to each other. In particular, I will show how this relationship is used to construct a combinatorial graph, called a Bratteli diagram, which can be used to encode the dimensions of the irreducible representations. I'll show how this graph is constructed for several examples of pairs of algebras and, in particular, the Tenabe algebra. I'll show how it is related to set valued tableaux and I'll talk about the problem of finding a tableau model for the irreducible Tenabe modules and the goal of constructing the irreducible representations with the tableaux as a basis.Oct 18 Reading Week No Seminar Oct 25 Aram Dermenjian (York) Using flag f-vectors A flag $f$-vector is a sort of refining of the $f$-vector of a convex polytope $P$ which gives you more information about the facial structure of $P$. We go over the definition of a flag $f$-vector and show how to compute the flag $f$-vector for the Coxeter complex using descents. We then review the topics of even signed permutations and covectors in projective space. If we have time we'll show a bijection between even signed permutations with a single descent and covectors in projective space and extend this bijection to certain parabolic cosets in the type $B_n$ Coxeter group with certain chains of covectors.Nov 1 Robin Sulzgruber (York) Statistics on Coxeter complexes We will discuss the relation between the Coxeter complex of a Weyl group $W$ and the action of $W$ on a suitable quotient $R$ of the ring of super-polynomials. The elements of the Coxeter complex can be viewed as certain ordered set partitions. Our goal is to find combinatorial statistics on these objects that realise the Hilbert series of the graded representation $R$.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 John Machacek (York) Partitionability and sign variation We will discuss an approach to partitioning simplicial complexes coming from sign variation. A map from chains in the face poset to signed permutations will be used.Nov 22 Robin Sulzgruber (York) Graßmann algebras, Gröbner bases, and dihedral groups We will review the theory of Gröbner bases for algebras that contain both a symmetric algebra and an exterior algebra. Our goal is to apply Gröbner bases to super-invariants of dihedral groups.Nov 29 John Machacek (York) Linear super harmonics (and beyond?) We will describe the super harmonics of degree 1 in the commuting $x$-variables. This will establish the coefficient of $qz^k$ in the Hilbert series for each $k$. Possible approaches to higher degrees as well as to describing the symmetric group representation will also be given.dec 6

Date Speaker Title (click titles for abstract) Jan 10 Jan 17 Jan 24 Jan 31 Aram Dermenjian (York) Twisting Bruhat to find weak order joins We introduce a preclosure on sets of reflections called the Bruhat preclosure which is defined by a twisting of the Bruhat graph of a Coxeter group. By a conjecture of Dyer; this preclosure is another method of finding the join in the weak order of Coxeter groups using the inversion set. This conjecture has been solved for dihedral groups and for the symmetric group, but is still open for all other finite Coxeter groups. We go over this conjecture and its implications in infinite Coxeter groups.Feb 7 Muqiao Huang (York) Weak composition quasi-symmetric functions A hopf algebra of weak compositions WCQSYM is defined, using power series with exponent being not necessarily the natural numbers, certain semi groups. Despite living in an uncommon space, the ordinary QSYM is actually a quotient of WCQSYM. More interestingly, there is a way of seeing WCQSYM as the free object on one generator in the (unheard of before, to me) category of Rota-Baxter algebras.Feb 14 Robin Sulzgruber (York) LLT polynomials and bounce paths LLT polynomials are symmetric functions with a tendency to emerge in the context of difficult combinatorial problems. We will discuss linear relations between certain LLT polynomials in conjunction with a combinatorial construction called the bounce path. In special cases these relations are relevant to the open problem of expressing LLT polynomials in terms of k-Schur functions.Feb 21 NO SEMINAR Reading Week Feb 28 Justin Troyka (York) Foulkes' conjecture and its generalizations The long-standing Foulkes' Conjecture asserts that a certain difference of plethythms of complete homogeneous symmetric functions is Schur-positive. Vessenes has stated a Generalized Foulkes' Conjecture. I will explain these two conjectures and some work done on them. I will also introduce my own generalization, which I have reason to believe is more natural and perhaps easier to prove. All of these conjectures about symmetric functions specialize to conjectures about q-binomial coefficients, which have a nice combinatorial interpretation.Mar 6 Nantel Bergeron Summary of conjectures on super-harmonics This weeks talk will be a collaborative effort. Our goal is to recall, review, and find a concise way to write down our results and conjectures on super-harmonics.Mar 14 Robin Sulzgruber Recap of the cycling map This weeks we will continue our recap of our results and conjectures on super-harmonics. In particular, I will explain the "cycling map" which, conjecturally, yields a basis of the space of super-harmonics with desirable properties.Mar 20 Robin Sulzgruber Positivity of differences of q-binomials (online) This week we will revisit some of the conjectures presented by Justin on the positivity of the differences of certain (q-)binomial coefficients, the implications for lattice paths, and possible generalisationf to the world of Fibonacci numbers. Everyone is welcome to contribute to this discussion.Mar 27 Justin Troyka Positivity of differences of q-binomials (online) This week we will continue our discussion of certain positivity conjectures concerning q-binomial coefficients and q-fibonomials. Everyone is welcome to contribute questions or ideas.Apr 3 Mike Zabrocki Super harmonics and super coinvariants (online) Three comments on the super-harmonics/coinvariants:

1. The definition of a set of generators that we can try to use to extract a monomial basis for our space.

2. Yet another conjectured basis for the super-harmonics/coinvariants that uses the Bergeron-Garsia basis for the Hall-Littlewood module.

3. State a recurrence that we should observe by restriction from $S_n$ to $S_{n-1}$.Apr 10

- 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.