The Algebraic Combinatorics Workgroup

Scheduled for Fridays at 3:15 PM (EDT [GTM -4]),

#### Schedule FALL 2022
Click to see/hide schedule

#### Schedule WINTER 2023
Click to see/hide schedule

#### Schedule SUMMER 2023
Click to see/hide schedule

A working seminar (at) of the Fields Institute organized by
Nantel Bergeron
and Kelvin Chan.

Date Speaker Title (click titles for abstract) Sept 2 Lucas Gagnon (York), Anthony Aurthur Lazzeroni Jr (HKBU), Farhad Soltani (York), Nantel Bergeron (York) First Meeting. Icebreaker and Problem Suggestions A few people will present some icebreaker talks and suggest some open problems.Sept 9 Farhad Soltani (York) DQSYM I will be talking about quasisymmetric polynomials on two sets of variables and the coinvariant space.Sept 16 Lucas Gagnon (York) Chromatic quasisymmetric functions and representation theory This talk will explore several topics related to the chromatic symmetric function and its quasisymmetric q-analogue. First, I will use them as a way to reviewt (quasi-)symmetric functions and some of their properties. Then, I will talk about some representation theoretic interpretations for symmetric functions generally, and the chromatic quasisymmetric function in particular.Sept 23 Anthony Aurthur Lazzeroni Jr (HKBU) The ideal of r-quasisymmetric functions In this talk I will recall the space of r-quasisymmetric functions. These spaces are nested subHopf algebras that interpolates between the space of Quasisymmetric functions and Symmetric functions. I will also discus some previous results about the quotient ring of the polynomial ring and the ideal generated by non-constant homogenous quasisymmetric functions.Sept 30 Alan Lao (York) The matroid represented by adjacency matrix of a graph The polygon matroids (or cycle matroids) of graphs have been extensively studied. I am going to introduce parts of the works by Brijder R., Hoogeboom H. J. and Traldi L. on the adjacency matroids of graphs. In particular we will look at the effect of "loop pivoting" on the adjacency matroids, and we will also see that "loop pivoting" can be seen as an action on a system which is another way of representing a graph.Oct 7 Nantel Bergeron (York) Polytopes and Gröbner theory We will review some background on polytopes and Gröbner theory. Then we will collectively explore using Gröbner theory to study properties of polytopes.Oct 14 N/A Cancelled (reading week) Oct 21 Kelvin Chan (York) Super harmonics Super harmonics are anticommuting variants of diagonal harmonics introduced in this seminar series back in 2018. We will trace a little bit of its history and talk about some progress and open problems.Oct 28 Nancy Wallace (York) Decomposing Diagonal Harmonics into GL_2\times S_n bicharacters We will see what this means, point out some open problems related to this and talk about some progress. We will end by explaining the relation between diagonal harmonics and Shi arrangements.Nov 4 Nantel Bergeron (York) QSym varities continued We will continue our discussion on QSym varities from 3 weeks ago.Nov 11 Anthony Aurthur Lazzeroni Jr (HKBU), Lucas Gagnon (York) Group effort! We will have a group talk today. Various people will discuss progress of their projects.Nov 18 Yohana Solomon (York) Hopf Algebra of Permutations We will review the combinatorial description of the Malvenuto-Reutenauer Hopf algebra of permutations, and define a new basis for this algebra.Nov 25 Nantel Bergeron (York) QSym varieties continued continued TBDDec 2 N/A Cancelled due to conflict with CMS Winter meeting. None.Dec 9 Anthony Aurthur Lazzeroni Jr (HKBU) r-QSym coinvariants continued See title. :)Dec 16 Kelvin Chan (York) A potential leading term argument for super harmonics We will examine some data that suggests there is a leading term argument for the conjectured basis of super harmonics.

Date Speaker Title (click titles for abstract) Jan 27 Curran McConnell (York) Weingarten Functions and Second-Order Finite Free Probability Finite free probability is a field concerned with the eigenvalues of random matrices, especially under the operation of matrix addition. The matrix A+UBU* is of particular importance, where U is a Haar-distributed random unitary matrix, and A and B are deterministic matrices. The "first-order" behaviour of this random matrix's characteristic polynomial, i.e., its expected value, is well understood. I am working to develop the theory of its second-order behaviour, i.e., its covariance matrix. Narrowing in on the variance of the determinant of A+UBU*, I will describe how Weingarten functions connect Haar unitary matrices with the representation theory of the symmetric group.Feb 3 Nantel Bergeron (York) Everything you want to know about quasisymmetric varieties, excedance classes and Temperley-Lieb algebra Another installment of the critically accliamed “QSV” series. ;)Feb 10 Kelvin Chan (York) Recent development in superspace coinvariants We survey a recent paper posted to arxiv on the Hilbert series of superspace coinvariants.Feb 17 Lucas Gagnon (York) Things you didn’t want to know about excedance classes In a continuation of Nantel’s talk, I will say (and prove) even more about excedance classes and the set QSV, including some new statements and open problems about the weak order.Feb 24 N/A Cancelled (reading week) Mar 3 Mike Zabrocki (York) Diagram algebras and representation theory I will talk about the representation theory of diagram algebras and how it relates to the representation theory of the symmetric group. I’ll explain how it relates to some open problems in combinatorial representation theory.Mar 10 Félix Gélinas (UofT) Cones and ping-pong in three dimensions My team and I studied the hypergeometric group in GL_3(CC) with parameters α=(1/4,1/2,3/4) and β=(0,0,0). In this talk I give a new proof that this group is isomorphic to the free product Z/4Z∗Z/2Z by exhibiting a ping-pong table. This table is determined by a simplicial cone in R^3, and I will prove that this is the unique simplicial cone (up to sign) for which this construction produces a valid ping-pong table. This talk is based on joint work with Gabriel Frieden and Étienne Soucis.Mar 17 Kelvin Chan (York) A survey of Kronecker rule for one hook shape Following up on Mike’s talk, we will go over a simplified proof of the Kronecker rule for one hook shape due to Ricky Liu.Mar 24 Farhad Soltani (York) A conjectured combinatorial interpretation of the dimension of the diagonally quasisymmetric polynomials coinvariant space See title.Mar 31 Mike Zabrocki (York) From stable to unstable Kronecker coefficients The coefficient of a Schur function s_la in the Kronecker product s_mu * s_nu are denoted g_(la,mu,nu) and are known as the Kronecker coefficients. For n sufficiently large, as n increases the sequence of coefficients g_((n-|la|,la)(n-|mu|,mu)(n-|nu|,nu)) are known to weakly increase and then stabilize. These limits for large n are sometimes referred to as stable Kronecker coefficients. I will explain a problem related to recovering a combinatorial interpretation for the coefficients g_(la,mu,nu) from a combinatorial interpretation for the stable Kronecker coefficients.Apr 7 N/A Cancelled (Good Friday) Apr 14 N/A Apr 21 Nancy Wallace (York) Decomposing Diagonal Harmonics into GL_2\times S_n bicharacters, the other side of the story Last time we talked about the GL_2 side of the partial formula for GL_2\times S_n bicharacters of diagonal harmonics. This time we will discuss the S_n side, and introduce some Schröder-like words.

t

Date Speaker Title (click titles for abstract) May 12 Lucas Gagnon (York) Gradings for diagram algebras We will examine some combinatorial ways to decompose the regular representation of a diagram algebra. This approach generates a few natural filtrations of the regular module that are relevant to ongoing work.May 19 Curran McConnel (York) An approach to counting the symmetric group's p-Sylow double cosets Double cosets are worth studying due to their interesting statistical interpretations, but the double cosets LS_nL of the symmetric group with respect to a Sylow p-subgroup L are not yet well-understood. Even determining the cardinality of LS_nL is an open problem. I will share a novel approach to indexing these double cosets, using families of antisymmetric matrices with certain properties. This approach is currently restricted to the case n = p^a, and is still conjectural. But the main conjectures have some empirical validation in the form of computerized tests. I will describe the conjectured index set, then outline how one would use it in order to determine the cardinality of LS_nL.June 2 Kelvin Chan (York) Polarization equivalence in superspace We (re)introduce polarization equivalence and prove certain two sets of symmetric operators are polarization equivalent in superspace.

- This is in collaboration with 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.