The Algebraic Combinatorics Seminar

Scheduled for Fridays at 3:30 PM, 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 28 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.

Robin:P-partitions and Schur-positivity: Omega of Chromatic symmetric functions and omega LLT have some positivity properties that we want to study. it is known that they are $K_p$ positive (where $K_p$ is a special kind of P-partition for a poset p). Question: When is a positive linear combination of $K_P$ Schur positive or $h$-positive?. [add references]

John:Fibolan Returns: In 2013-14 this seminar studied a lot replacing $n$ by the $n$th Fibonacci number in known formulas (like binomial, Catalan number, etc). We also considered q-version of Fibonacci number. In 2013-14 we have a combinatorial model for q-Fibo-binomials, but the model didn't allow to understand well (conjectured) positivity of q-Fibo-Catalan. John presented a new combinatorial model that may allow us to solve the q-Fibo-Catalan positivity conjecture. [see this paper of Bennett-Carrillo-M.-Sagan]

Mike:Super Coinvariants: Last year, we introduces supersymmetric polynomials. We want to consider the quotient the super space by the augmented ideal of supersymmetric polynomials. We call that quotient $SC_n$ when we have $n$ commuting variables and $n$ anticommuting super variables.

CONJ 1: dimension of $SC_n$ is the number of preferential arrangements set compositions

CONJ 2: q-t-dimension is a sum with q-stirling number and the number of alternants is a q-y-analogue of $2^{n-1}$.

CONJ 3: We have a formula for the full Frobenius.

[add reference]

Nantel:The steep-bounce conjecture: With Pilaud and Ceballos we have conjectured that the set of pair of nested Catalan paths $(\pi_1,\pi_2)$ where $\pi_1$ is a pure bounce path with $k+1$ touches on the diagonal is in bijection with the set of nested Catalan paths $(\pi'_1,\pi'_2)$ where $\pi'_2$ is a steep path with $n-k$ steps. [see this paper of B-Pilaud-Ceballos]Oct 5 R Sulzgruber (York) Zeta/sweep map I will provide some background on the steep-bounce conjecture presented by Nantel last week. In particular I will introduce the zeta map (also called sweep map), which is a useful bijection in the theory of Dyck paths and diagonal harmonics.Oct 12 Reading Week No Seminar Oct 19 Nantel Bergeron (York) q-Fibolan positivity conjecture Oct 26 John Machacek (York) Gröbner basis theory, coinvariants, and superspace We will recall some Gröbner basis theory and how it applies to classical coinvariants. We will then try to extend to superspace.Nov 2 Alexander Nenashev (York) An approach to the question on dotted orders stated by Nantel last year Consider a linear order on $[n]=\{1,\dots,n\}$ with dots between some of the consecutive numbers in this order; examples $35.7.241.6$ or $62.175.43$ ($n = 7$). I call such a structure a dotted order on $[n]$ and denote it $(L,\lambda)$, where $L$ is a linear order and $\lambda$ is a subset of $\{1,\dots,n-1\}$. Given two orders $L$ and $M$ on $[n]$, denote $d(L,M)$ the choice of points showing the descents of $M$ with respect to $L$; example: $M = 235164$ has descents $23.5.1.64$ with respect to $L = 641253$, and $L$ has descents $64.1.25.3$ with respect to $M$. Let $A_n$ be the free abelian group on the dotted orders on $[n]$; its rank is clearly equal to $n!2^{n-1}$. Now consider the endomorphism $\phi_n$ of $A_n$ which sends each generator $(L, \lambda)$ to the sum of all $(M, \mu)$ such that $d(L,M) = \mu$ and $d(M,L) = \lambda$. Last year Nantel Bergeron raised the question to understand $\phi_n$ and to determine its kernel in particular. The case $n=3$ is easy, however there is a question that I cannot answer already in this case. And the purpose of my presentation is to speak on how I look at the $n=4$ case in terms of a certain graph and to state questions arising from it.Nov 9 Shu Xiao Li (York) Techniques for proving $e$-positivity and Schur-positivity I will provide some background about results and techniques used to prove $e$-positivity and Schur-positivity of certain classes of symmetric functions. And then we try to translate it to $K_p$ i.e. find classes of posets such that the sum of the $K_p$ is e/Schur-positive.Nov 16 John Machacek (York) Some more terms and a new formulation for super coinvariants We will discuss more leading terms and standard monomials for super coinvariants. These terms and monomials come from patterns which have been found in the data leading to an increase both in our proven understanding as well as our conjectural understanding. An alternative recursive version of the Hilbert series will also be presented.Nov 23 Robin Sulzgruber (York) Dual equivalence graphs Dual equivalence graphs were introduced by Sami Assaf as a tool for proving Schur-positivity of symmetric functions. We will recall their definition and some basic properties.Nov 30 Shu Xiao Li (York) Coinvariants in super space I will give a summary about our recent results on the coinvariant space of symmetric functions in super space. In particular, we try to find a suitable monomial order and construct a linear basis in the quotient. I will give a brief review on the Aval-Bergon-Bergeron proof of $dim(R[x_1,\dots,x_n]/\langle QSym^+\rangle) = Catalan_n$, and we try to extend their technique for our super coninvariants and set-compositions.

Date Speaker Title (click titles for abstract) Jan 11 Nantel Bergeron (York) Harmonics and coinvariants are equivalent in superspace... and more. I will go through Mike's email and show what I can. This is a good way to get back to the problems at hands.Jan 18 Kelvin Chan (York) The Steep-Bounce Theorem The Steep-Bounce Conjecture was recently resolved by Ceballos, Fang and Mühle. We will take a look at their bijective proof.Jan 25 NO SEMINAR Banf Workshop and CAAC conference in Ottawa Feb 1 Changjian Su (University of Toronto) Motivic Chern classes and Iwahori invariants of principal series Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump-Nakasuji-Naruse about certain transition matrix between these two bases. The idea of the proof is to use the two geometric realizations of the affine Hecke algebra, and relate the Iwahori invariants to Maulik-Okounkov's stable envelopes and Brasselet-Schurmann-Yokura's motivic Chern classes for the Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.Feb 8 Mike Zabrocki and John Machacek (York) Symmetric and harmonic polynomials in superspace We will discuss the quotient $\mathbb{Q}[X_n,\Theta_n]/I^+$ where $I^+$ is the ideal generated by symmetric polynomials in superspace with vanishing constant term. We will present new conjectures that involve more sets of variables. (slides and Sage worksheet)Feb 15 Shu Xiao Li (York) Towards a basis for super-harmonics We will discuss a lower bound for the Hilbert-series of harmonic polynomials in super-space. In particular we will try to make progress in proving such a lower bound.Feb 22 NO SEMINAR Reading Week Mar 1 Robin Sulzgruber (York) Parking functions and ribbon Schur functions We will review the necessary combinatorics of parking functions and related objects to understand the following new question posed by Nantel: Can we expand the shuffle ex-conjecture in terms of ribbon Schur functions? This question is related to the steep-bounce ex-conjecture.Mar 8 Mike Zabrocki Garnir polynomials and super-harmonics We conjecture that the space of super harmonics has a positive expansion in terms of Hall-Littlewood symmetric functions twisted with a sign character and an expansion in terms of $\omega C_\alpha[X;1/q]$ functions with $2^{n-1}$ terms. I will give an introduction to Garnir polynomials and "higher Specht polynomials" and their relation to Hall-Littlewood symmetric functions (see "On certain spaces of Harmonic Polynomials" N. Bergeron and Garsia and "Higher Specht polynomials for the symmetric group" by Terasoma and Yamada) so that hopefully we can exhibit this decomposition explicitly.Mar 15 Nantel Bergeron and John Machacek Garnir polynomials as operators on alternants We will continue from last week and discuss applying Garnir polynomials as operators on alternants obtained by polarization in the super harmonics.

- 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.