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.

#### Schedule FALL 2018 Click to see/hide schedule

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.

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
Nov 30

Notes.

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

Year Topic
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