Abstracts

Martina Balagović

Finite dimensional representations of quantum symmetric pair coideal subalgebras of type BII

Quantum symmetric pair coideal subalgebras of type $BII_N$ are pairs $\mathcal{B}_c\subseteq U_q \mathfrak{so}_{2N+1}$ deforming classical symmetric pairs $U\mathfrak{so}_{2N}\subseteq U \mathfrak{so}_{2N+1}$. We classify finite dimensional representations of $\mathcal{B}_c$ over a wide choice of fields with $q$ not a root of unity. Specifically, we describe the Letzter-Cartan subalgebra of $\mathcal{B}_c$, specify fields over which all finite dimensional representations of $\mathcal{B}_c$ are highest weight representations, and describe the notion of "integral" highest weights of the Letzter-Cartan subalgebra which correspond to finite dimensional representations of $\mathcal{B}_c$. This is joint work with Stefan Kolb and Xinyang Liu.

Elijah Bodish

Type B webs

This talk is about joint work with Elias–Rose on a diagrammatic presentation of the monoidal category of type B fundamental representations. Our approach parallels Cautis–Kamnitzer–Morrison's skew Howe duality approach to type A webs, using Wenzl's quantum spin Howe duality in place of quantum skew Howe duality. I will emphasize connections to: $\imath$quantum Weyl group symmetries of nonclassical modules, and our previous work on folded categorified skew Howe duality.

Jon Brundan

Yangians and degenerate affine Schur algebras

I'll talk about some things I proved 20 years ago which I understand better now, and some things Drinfeld proved 40 years ago which I still do not understand. This talk is based on my joint work with Slava Ivanov (see arXiv:2512.04253 for more sensible abstract). Our work was based in part on another recent paper by Song and Wang (see arXiv:2406.13172).

Nicolle González

Title and abstract to be announced.

David Green

Multifusion Hopf monoidal categories are group theoretical

I will review the definition of a Hopf multifusion category, and present a couple results, some of which are joint with Brett Hungar and Sean Sanford. The main result, which is work in progress, leverages the Kac exact sequence to provide (all) the first nontrivial examples of these structures since the pioneering work in the late 90's.

Mee Seong Im

Diagrammatics of entropy

I will discuss how cocycles appear in a graphical network. Furthermore, the Shannon entropy of a finite probability distribution has a natural interpretation in terms of diagrammatics. I will explain the diagrammatics and their connections to infinitesimal dilogarithms and entropy. If there is time, I will discuss how algebraic K-theory is related to cobordism groups of foams in various dimensions. This is joint with Mikhail Khovanov.

Stefan Kolb

Short star products for quantum symmetric pairs

Short star products are filtered quantizations of graded algebras satisfying a truncation condition first considered by Beem–Peelaers–Rastelli and further developed by Etingof and Stryker. In this talk I will explain that quantum symmetric pair coideal subalgebras are realized as short star products on quantum horospherical subalgebras. The shortness property allows for immediate conceptual interpretations of antiautomorphisms and bar-involutions which had previously been constructed via the quasi $K$-matrix. Moreover, this perspective allows us to express the quasi $K$-matrix in terms of the quasi $R$-matrix of Drinfeld and Lusztig. The talk is based on joint work with Milen Yakimov.

Melody Molander

NIM-representations of Tambara–Yamagami generalizations

Tambara–Yamagami (TY) categories form one of the simplest families of non-pointed fusion categories. Despite their elementary description, these categories play a central role in mathematical physics. Studying generalizations of TY fusion rings can then provide a controlled framework to explore the new phenomena that arise when one moves beyond the classical TY setting. In this talk, I will focus on the generalization of the TY fusion ring, proposed by Jordan–Larson. One of the most useful tools to study fusion rings is the classification of their non-negative integer matrix (NIM-) representations. NIM-reps can be used as an effective method to detect algebra objects in the fusion category underlying the fusion ring. In this talk, we will compute and classify the irreducible NIM-reps of both proposed extensions as well as detect candidate algebra objects associated to these NIM-reps. This work is joint with Agustina Czenky, Emily McGovern, Monique Müller, and Ana Ros Camacho.

Khoa Nguyen

On $U(\mathfrak h)$-free $\mathfrak{sl}_2$-modules of finite rank

In this talk, we discuss non-weight $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ denotes the Cartan subalgebra of $\mathfrak{sl}(2)$. In particular, I will present a classification of simple scalar-type $U(\mathfrak h)$-free modules of rank $2$ and introduce several families of simple finite-rank $U(\mathfrak h)$-free modules. This talk is based on joint work with D. Grantcharov and K. Zhao.

Jianping Pan

Uncrowding algorithms and crystal structures in K-theoretic combinatorics

Crystal bases provide powerful combinatorial models for representations of quantum groups, with semistandard Young tableaux being the classical type A example. K-theoretic Schubert calculus gives rise to richer families of tableaux, including set-valued and hook-valued tableaux, whose generating functions are stable Grothendieck polynomials and their canonical analogues. In this talk, I will describe how these K-theoretic tableau models can be understood through crystal-theoretic methods. A key tool for understanding these objects is an "uncrowding" operator, which relates more complicated tableaux back to classical ones while preserving structural features. Building on Buch's uncrowding operator for set-valued tableaux, I will discuss a recent extension to hook-valued tableaux. By studying these operators, we uncover a "hidden" symmetry of hook-valued tableaux through jeu de taquin, using a classical result of Benkart, Sottile, and Stroomer. This talk is based on joint work with Jang, Kim, Pappe, Poh, and Schilling.

Shifra Reif

The Harish-Chandra theorem for symmetric superspaces and ghost distributions

The classical Harish-Chandra theorem describes the center of the universal enveloping algebra via Weyl-group-invariant polynomials on the Cartan subalgebra. This theorem admits two natural generalizations: one to invariant differential operators on symmetric spaces and one to Lie superalgebras. In the super-setting, we can also define the anti-center which contains certain square roots of central elements. Its Harish-Chandra image was computed by Gorelik and the construction was generalized to symmetric superspaces by Sherman, linking them to certain invariant "ghost" distributions.

In this talk, we will combine these generalizations and describe the Harish-Chandra theorem for symmetric superspaces as well as for ghost distributions.

Joint work with Siddhartha Sahi, Vera Serganova and Alexander Sherman.

Siddhartha Sahi

Title and abstract to be announced.

Vera Serganova

Balanced and neat elements in quasi-reductive Lie superalgebras

Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group is reductive). To understand better odd elements in quasi-reductive Lie superalgebras, we consider two trivially-intersecting classes of odd elements, which turn out to be particularly convenient to work with: neat odd elements and balanced odd elements. Neat elements are always ad-nilpotent, and are the best analogues one can consider for nilpotent elements in a semisimple Lie algebra, since they may be embedded into subalgebras which are isomorphic to $\mathfrak{osp}(1|2)$, a simple Lie superalgebra whose underlying Lie algebra is $\mathfrak{sl}(2)$. Balanced odd elements, on the other hand, are natural generalization of the notion of a self-commuting element (an odd element such that $[x, x] = 0$), and are used to define homology-type functors on the category of representations of $G$. We study the properties of balanced and neat odd elements in quasi-reductive Lie superalgebras, and show that any odd element $x$ may be written as a sum of a neat and a balanced odd element which commute with each other.

This theorem has a very nice categorical application. Let $\mathfrak{g}(1|1)$ be the $(1|1)$-dimensional Lie superalgebra generated by an odd element $x$. The semisimplification of the category of finite-dimensional super-representations of $\mathfrak{g}(1|1)$ is a symmetric monoidal functor $S \colon \operatorname{Rep}(\mathfrak{g}(1|1)) \to \operatorname{Rep}(\mathrm{OSp}(1|2))$. Any odd $x$ in $\operatorname{Lie}(G)$ induces a homomorphism $i \colon \mathfrak{g}(1|1) \to \operatorname{Lie}(G)$. Let $\Phi(x) \colon \operatorname{Rep}(G) \to \operatorname{Rep}(\mathrm{OSp}(1|2))$ be the composition of the restriction functor and the semisimplification functor $S$. We show that the functor $\Phi(x)$ may be described explicitly using the homology-type functor corresponding to the balanced part of $x$ in the above decomposition. These homology-type functors are known as Duflo–Serganova functors.

The talk is based on joint work with Inna Entova-Aizenbud.

Weiqiang Wang

Categorifying quantum affine $\mathfrak{gl}_p$ and its integrable modules

Cyclotomic $q$-web categories (introduced jointly with Yaolong Shen and Linliang Song) have produced new Schur algebras sitting in between cyclotomic Hecke algebras and cyclotomic $q$-Schur algebras. We will explain that a module category over a cyclotomic $q$-web category for $q$ a root of unity categorifies an integrable highest weight module over quantum affine $\mathfrak{gl}_p$, and the projective indecomposable modules categorify the canonical basis. To that end, we connect the affine $q$-webs to Hall algebra of the cyclic quiver and to geometric representation theory. This generalizes the classic works of Lascoux–Leclerc–Thibon, Ariki, and Varagnolo–Vasserot. Based on joint work with Linliang Song.

Curtis Wendlandt

Quantum vertex structures and quantum groups

The notion of a vertex algebra is closely related to that of a commutative algebra. The latter provide the simplest examples of the former and, conversely, a general vertex algebra can be understood as a singular commutative algebra whose multiplication map depends on a parameter. This relation admits a natural quantization where the notion of a vertex algebra is replaced by that of quantum vertex algebra, which was introduced by Etingof and Kazhdan over 25 years ago. In this quantized story, commutative algebras grow up to become almost commutative algebras in the sense of quantum groups — their product and opposite product are intertwined by a solution of the quantum Yang–Baxter equation. The goal of this talk is to expand on this connection by explaining how to construct quantum vertex algebras within the category of representations of a distinguished class of quantum groups. This construction is a generalization of Majid's transmutation theory for quasitriangular Hopf algebras, and comes with a functor from the category of representations of the underlying quantum group to the category of modules over the resulting quantum vertex algebra. When applied to the Yangian of a simple Lie algebra, this yields a uniform construction of the Etingof–Kazhdan quantum affine vertex algebra whose structure maps are given by closed formulas. This is based on joint work with Alex Weekes and Matt Rupert.

Weinan Zhang

Quantum symmetric pairs at roots of unity

The representation theory of quantum groups at roots of unity was developed by De Concini–Kac–Procesi, and this has applications and connections in modular representations, Poisson geometry, and geometric representation theory. In this talk, we generalize the approach of De Concini–Kac–Procesi to quantum symmetric pairs. We construct the Frobenius center for $\imath$quantum groups at odd roots of unity and show that the irreducible modules of $\imath$quantum groups are parametrized by twisted conjugacy classes of the underlying Lie group. We further study dimensions of these irreducible modules. This is based on a joint work with Jinfeng Song.