Date	Speaker	Title (click titles for abstract)
9 Dec. 2024	Stephan Pfannerer-Mittas U. Waterloo	Rotation-invariant web bases from hourglass plabic graphs In 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of U_q(sl_3). Extending these bases to general sl_r has been an open problem ever since. We present a solution to the r=4 case by introducing hourglass plabic graphs - a new generalization of Postnikov's plabic graphs. Joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson.
2 Dec. 2024	Adrien Segovia	The free and parking quasi-symmetrizing actions This talk is about combinatorial Hopf algebras. Some of these algebras have equivalent definitions as the invariants under a symmetric group action. Such an example is given by the quasi-symmetrizing action of Hivert, which recovers the Hopf algebra QSym. We will give two new examples: the free quasi-symmetrizing action for FQSym and the parking quasi-symmetrizing action for PQSym*. In the case of PQSym*, we generalize our action and obtain an infinite chain of nested graded Hopf subalgebras, which we study combinatorially.
25 Nov. 2024	Ravali Nookala	On Artin-Schelter Regular Algebras of Dimension Four In this talk, I will present two interrelated aspects of Hochschild cohomology for Koszul Artin-Schelter regular algebras of dimension four. First, I will discuss the computation of Hochschild cohomology and the Kodaira-Spencer map for known families of these algebras. I will highlight how the surjectivity and bijectivity of the Kodaira-Spencer map relate to the structure of the moduli space. Second, I will examine the Hochschild cohomology of the Koszul dual of k_Q[x_1,…,x_4] and our goal to focus on the influence of certain parameters on the cohomology and the construction of families that intersect the k_Q[x_i] component. This research sheds light on the algebraic and geometric properties of these regular algebras and their moduli spaces.
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11 Nov. 2024	break	TBA
4 Nov. 2024	Antoine Abram UQAM	Power quotients of plactic-like monoids The genesis of this project is the stylic monoid. This monoid can be describe as a quotient of the plactic monoid by the relations x^2=x. Inspired by the nice properties of this monoid, we considered more general quotients, power quotients. For these quotients we let the exponent in our relation be any integer >1, which can differ from one letter to another. We studied power quotients of particular monoids called the plactic-like monoids. Plactic-like monoids are monoids that can defined by means of an insertion algorithm in some combinatorial objects. In this talk we will consider three monoids: the plactic monoid itself, the Chinese monoid and the Sylvester monoid. In all these cases, we were able to fully understand the structure of any power quotient by understanding the 2-quotient; i.e. the power quotient where the exponents are 2. These 2-quotient have nice properties, for instance their enumeration uses classical sequences of integers. Based on a joint work with F. Hivert, J. Mitchell, J.-C. Novelli and M. Tsalakou
28 Oct. 2024	Aaron Lauve Loyola Chicago	Adjoint modalities in combinatorial Hopf algebras In studying Hopf algebras $H=\bigoplus_{n\geq0} \mathbb{Z}P_n$ built on combinatorial gadgets $P_\bullet$ with a natural poset structure, it is natural (and often fruitful) to use the poset to define new bases for $H$. Here I introduce a variation on the common technique, when in the presence of poset maps $f_n:P_n \to Q_n$. After introducing the basic framework, I'll give lots of examples and a couple consequences. As the title suggests, the former requires the maps $f_n$ to be part of an *adjoint modality.* Based on joint work with Marcelo Aguiar. (In progress.)
21 Oct. 2024 Special time: 3:35 PM	Lucas Gagnon York	Equivariant forest polynomials and the quasisymmetric flag variety Schubert calculus studies intersections inside the flag variety $\mathrm{GL}_{n}/B$ by realizing cohomology classes as elements of the polynomial ring $R_{n} = \mathbb{C}[x_{1}, \ldots, x_{n}]$ modulo symmetric polynomials. Schubert polynomials give a beautiful and concise way of computing this realization. Building on the framework of Schubert polynomials, this talk explores a generalization to quasisymmetric polynomials and Catalan combinatorics. I will identify a subvariety of $\mathrm{GL}_{n}/B$ whose cohomology is isomorphic to polynomials modulo quasisymmetric polynomials and exhibit this isomorphism using a family of Schubert-like polynomials indexed by binary forests. Finally, I will show how extending the story to equivariant cohomology frames these “forest” polynomials as the solution of a natural interpolation problem on noncrossing partitions. Based on joint work with Nantel Bergeron, Philippe Nadeau, Hunter Spink, and Vasu Tewari.
	Reading Week
7 Oct. 2024	Jerónimo Valencia U. Waterloo	A combinatorial proof of an identity involving Eulerian numbers In 2009, Brenti and Welker studied the Veronese construction for formal power series which was motivated by the corresponding construction for graded algebras. As a corollary of their algebraic computations, they discovered an identity for the coefficients of the Eulerian polynomials. The authors asked for a combinatorial proof of this identity given that all of its ingredients are enumerative in nature. In this talk I will present one such combinatorial proof.
30 Sept. 2024	Nantel Bergeron York	A quantum Murnaghan--Nakayama rule for the flag manifold Based on joint work with Benedetti, Colmenarejo, Saliola, and Sottile (arxiv:2406.053311). We give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung--Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.
23 Sept. 2024	Hugh Thomas UQAM	Cyclic actions on noncrossing and nonnesting partitions Noncrossing partitions and nonnesting partitions are both counted by Catalan numbers. Noncrossing partitions on [n] admit a natural cyclic action of order 2n, induced by the Kreweras complement. Nonnesting partitions admit a natural toggle-based action; in fact, they admit one such action for each choice of Coxeter element of the symmetric group. We prove that the latter actions all have order 2n by constructing a family of bijections between noncrossing and nonnesting partitions, equivariant with respect to the cyclic actions on either side. This talk is based on arXiv:2212.14831, joint with Benjamin Dequêne, Gabriel Frieden, Alessandro Iraci, Florian Schreier-Aigner, and Nathan Williams. Our results were presented at FPSAC in July; our extended abstract (and Nathan's slides) are available from the FPSAC website.