The Algebraic Combinatorics Seminar

Scheduled for Fridays at 3:15 PM (EDT [GTM -4]) in Room 210.

#### Schedule FALL 2023
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#### Schedule WINTER 2024
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#### Schedule SUMMER 2024
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A working seminar (at) of the Fields Institute organized by
Nantel Bergeron,
Kelvin Chan
and Blessing Oni.

Date Speaker Title (click titles for abstract) Sep 8 Tianyi Yu (UC San Diego) Analogue of Fomin-Stanley algebra on bumpless pipedreams Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart's growth diagram.Sep 15 Icebreaker and Problem Suggestions Sep 22 Félix Gélinas (York) The volume polynomial of a Matroid To prove the Heron-Rota-Welsh conjecture, Petter Brändén and Jonathan Leake extended the notion of hereditary Lorentzian polynomials associated to simplicial complex to matroid theory. Denoting that such polynomials is the volume polynomial of the Chow ring of a matroid, it is interesting to apply a differential operator $D_u$ and exploring possible connections to the notion of partial derivatives of volume polynomials in geometry. For a given matroid M, the question arises: can we combinatorially describe $D_uVol_M$ when evaluated at specific elements?Sep 29 Kelvin Chan (York) A bijection that sends major index to cocharge We will discuss a bijection on standard tableaux that sends the major index to the cocharge statistics. We will also introduce the problem that motivated the search for this bijection.Oct 6 Lucas Gagnon (York) Quasisymmetric Varieties for Complex Reflection Groups This talk will consider Nantel’s suggestion to generalize quasisymmetric functions and varieties to other reflection groups in the context of classical invariant theory for these groups.Oct 13 N/A Cancelled (reading week) Oct 20 Farhad Soltani (York) DQSym We will discuss Quasisymmetric polynomials in 2 sets of variables which is called diagonally quasisymmetric polynomials. We will introduce a possible basis for the space quotient by the ideal generated by positive diagonally quasisymmetric polynomials.Oct 27 Vasu Tewari (UTM) Forest polynomials and harmonics for Qsym I will discuss properties of a new basis for the polynomial ring that refines Schubert polynomials and subsequently describe the dual story by considering volumes of combinatorial cubes. Joint work with Philippe Nadeau (Lyon and CNRS).Nov 3 Li, Yu (UofT) Some combinatorics about cluster algebras of finite type We explain how to construct a cluster algebra of finite type. The number of seeds in such a cluster algebra is the Catalan number. There is a fan, which I usually refer to as the cluster fan, whose maximal cones are in bijection with the seeds. We suggest how one might hope to construct Panyushev's conjectural involution on the set of ad-nilpotent ideals (whose cardinality is also the Catalan number) using Poincare duality on the cohomology of the toric variety corresponding to the cluster fan.Nov 10 Vasu Tewari (UTM) Forest polynomials and harmonics for Qsym (Part 2) We will pick up from where we left off two Fridays ago.Nov 17 Li, Yu (UofT) Some combinatorics about cluster algebras of finite type (Part 2) We will pick up where we left off two weeks ago.Nov 24 Sarah Brauner (LaCIM/UQAM) Card shuffling, q-analogues and derangements How many times do you need to shuffle a deck of cards to ensure it is adequately mixed? This is a question in probability theory, but for many methods of card shuffling, the answer relies on combinatorics and representation theory. In this talk, I will discuss several classical card-shuffling processes and introduce a natural q-deformation, which can be understood as a random walk on the (Type A) Hecke algebra. Motivated by questions of mixing times, I will present recent results and conjectures concerning the eigenvalues and eigenspaces of these (q-)shuffling operators. Along the way we will see derangements, desarrangements, and tableau combinatorics. This is joint work with Commins and Reiner, as well as Axelrod-Freed, Chiang, Commins and Lang.Dec 1 N/A Cancelled (CMS Winter meeting) Dec 8 Nantel Bergeron (York) cyclic action and forest volumes See title.

Date Speaker Title (click titles for abstract) Jan 12 N/A Cancelled Jan 19 Nantel Bergeron (York) Cyclic action on quasisymmetric quotient See title.Jan 26 N/A Cancelled (CAAC) Feb 2 Denys Bulavka (Charles University) A Hilton-Milner theorem for exterior algebras A set family F is pairwise-intersecting if every pair of its members intersect. In 1960, Erdős, Ko, and Rado gave an upper-bound on the size of a pairwise-intersecting family of k-sets coming from a ground set of size n. Moreover, they characterized the families achieving the upper-bound. These are families whose members all share exactly one element, so called trivial families. Later, Hilton and Milner provided the next best upper-bound for pairwise-intersecting families that are not trivial.

There are several generalizations of the above results. We will focus on the case when the set family is replaced with a subspace of the exterior algebra. In this scenario intersection is replaced with the wedge product, being pairwise-intersecting with self-annihilating and being trivial with being annihilated by a 1-form. Scott and Wilmer, and Woodroofe gave an upper-bound on the dimension of self-annihilating subspaces of the exterior algebra. In the current work we show that the better upper-bound coming from Hilton and Milner's theorem holds for non-trivial self-annihilating subspaces.

This talk is based on a joint work with Francesca Gandini and Russ Woodroofe.

Feb 9 Mike Zabrocki (York) Submonoids of the uniform block permutation algebra I will introduce a problem related to the representation theory of the symmetric group and the uniform block permutation algebra. I will introduce an unusual partial order on the partitions of k and a conjecture about the submonoids of the uniform block permutation algebra that contain the symmetric group.Feb 16 Lucas Gagnon (York) New statistics for quasisymmetric cycling I will talk about the character of the cyclic shift on quasisymmetric harmonics, including some new statistics that (conjecturally) describe this action and ways we might use these statistics to make progress on the overall question.Feb 23 N/A Cancelled (reading week) Mar 1 Mike Zabrocki (York) Submonoids of the uniform block permutation algebra I will give a follow-up talk to the question of finding the submonoids of the monoid uniform block permutation that contain the symmetric group. I will show a combinatorial construction of the order on partitions. I will also discuss implications for the more general problem of finding the submonoids of a monoid M that contain the group of units.Mar 8 Nantel Bergeron (York) Open problems on polytopes See title.Mar 15 Andy Wilson (Kennesaw State University) Problems in superspace "Superspace" is a generalization of the multivariate polynomial ring in which some variables commute (like bosons in mathematical physics) while others anti-commute (like fermions). Beginning with a conjecture by Mike Zabrocki in 2019, there has been much work on extending the combinatorics of the multivariate polynomial ring into superspace. In this informal talk, I will outline some recent successes of this program and many still-open problems.Mar 22 Yohana Solomon (York) Hopf algebra of set partition We will review the combinatorial description of the partition algebra, and define a new basis for this algebra to introduce the Hopf algebra of set partition.Mar 29 N/A Cancelled (Good Friday) Apr 5 N/A Cancelled (social) Apr 12 Antoine Abram (UQAM) Alcoved Signed Permutations We introduce a new partial order on signed permutations whose cover relations are determined by certain big ascents. Its Hasse diagram is dual to the alcove triangulation of the fundamental parallelepiped of the type C root system, as studied by Lam and Postnikov. This duality allows us to use Ehrhart theory to obtain a generating function for big ascents and, conversely, to combinatorially interpret the coefficients of h∗-polynomial of the type C half-open hypersimplices. We also obtain a relation between the distribution of covers in our poset and the usual descents in the “half” weak order of type BC. Moreover, we show that these family of posets converges, in a precise sense, to the lattice of strict partitions.

This is based on joint work with Jose Bastidas.

Date Speaker Title (click titles for abstract) May 10 Nantel Bergeron (York) Houston we have a problem! We may have 99 problems but the lack of research problems isn't one.May 17 N/A Cancelled May 24 Leo Jiang More on flow polytopes I will discuss some topics related to flow polytopes, with a view towards the first problem from Nantel’s talk two weeks ago.May 31 Felix Gelinas Gently discovering Gentle Algebras I will introduce Gentles algebra and the combinatorial perspective of its indecomposable representations. This talk is mostly inspired by Hugh Thomas mini-course notes from Montréal's ISM Discovery School 2024.Jun 7 N/A Cancelled (Workshop on Computational and Applied Enumerative Geometry) Jun 14 N/A Cancelled (no room) Jun 21 Jun 28

- 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).
- Photo taken on September 8, 2023

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