Schedule for: 19w5074 - Algebraic and Geometric Categorification

I will discuss recent progress towards the Harish-Chandra classification for simple modules in category $\mathcal{O}$ of rational Cherednik algebras, and in particular discuss the extent to which Hecke algebras know the answer to the problem. Based on joint work with D. Juteau.

Bernstein operators are vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. This correspondence bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture.

In this talk I will review the program of categorification of Verma modules of symmetrizable quantum Kac-Moody algebras and extend it to a categorification of tensor products of a Verma module and several integrable irreducibles. In the last part of the talk I will consider the case of $\mathfrak{sl}(2)$ and explain how its categorifications are endowed with an action of the Temperley-Lieb algebra of type B with two parameters. The material presented is based upon several collaborations with Grégoire Naisse, Ruslan Maksimau and Abel Lacabanne.

In this talk, I will explain how to define an exotic t-structure for coherent sheaves on a partial resolution of the nilpotent cone. Along the way, I'll mention the special cases of exotic sheaves on the Springer resolution and perverse coherent sheaves on the nilpotent cone, along with some of their applications to geometric representation theory. I will also discuss some results on the structure of the heart. This is joint work with Paul Sobaje and Kei Yuen Chan.

The Alexander polynomial for knots and links can be interpreted as a quantum knot invariant associated with the quantum group of the Lie superalgebra $\mathfrak{gl}(1|1)$. This polynomial has been famously categorified to a link homology theory, knot Floer homology, defined within the theory of Heegaard-Floer homology. Andy Manion showed that the Ozsvath-Szabo algebras used to efficiently compute knot Floer homology categorify certain tensor products of $\mathfrak{gl}(1|1)$ representations. For representation theorists, the work of Sartori provides a different categorification of these same tensors products using subquotients of BGG category $\mathcal{O}$. In this talk we will explain joint work with Andy Manion establishing a direct relationship between these two constructions. Given the radically different nature of these two constructions, transporting ideas between them provides a new perspective and allows for new results that would not have been apparent otherwise.

Let $G$ be a semisimple algebraic group over an algebraically closed field $F$ of very large positive characteristic. We give a combinatorial classification and find Kazhdan-Lusztig type character formulas for modules over the Lie algebra $\mathfrak{g}$ that are equivariantly irreducible with respect to an action of a certain subgroup of $G$ whose connected component is a torus. This is a joint work with Roman Bezrukavnikov.

Quantized Gieseker varieties are algebras that quantize functions on a Gieseker moduli space. Examples include algebras of differential operators on projective space and type A spherical rational Cherednik algebras. When a quantized Gieseker variety admits a finite-dimensional representation, it admits a unique simple one and it does not admit self-extensions. In my talk, I will define the quantized Gieseker variety as a quantum Hamiltonian reduction and explain how to explicitly construct its irreducible finite-dimensional representation from the irreducible finite-dimensional representations of the type A full rational Cherednik algebra. This is joint work with Pavel Etingof, Vasily Krylov and Ivan Losev.

Back in 2004, Rasmussen extracted a numerical invariant from Khovanov-Lee homology, and used it to give a new proof of Milnor's conjecture about the slice genus of torus knots. In this talk, I will describe a generalization of Rasmussen's invariant to null-homologous links in connected sums of $S^1 \times S^2$. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of null-homologous surfaces with boundary in the following four-manifolds: $B^2 \times S^2, S^1 \times B^3, CP^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. This is based on joint work with Marco Marengon, Sucharit Sarkar, and Mike Willis.

In this series of two talks we will explain the current state of the art concerning the problem of classifying "simple" 2-representations of Soergel bimodules. The first talk will introduce the asymtotic Hecke algebra and the asymptotic bicategory as well as their relationship. In particular, the asymptotic bicategories are well-understood in many cases, and this will be explained.

This is the sequel to the talk of Daniel Tubbenhauer. The second part will explain the 2-representation theoretic ingredients to reduce a classification of simple transitive 2-representations of Soergel bimodules in many (most) cases to the known classification of simple transitive 2-representations of the corresponding asymptotic bicategory.

The geometric Satake correspondence provides a beautiful geometric description of the representations of reductive groups, using the affine Grassmannian. In order to categorify this description, we use truncated shifted Yangians which quantize slices in the affine Grassmannian. Last year, we proved that category O for a truncated shifted Yangian is equivalent to a category of modules for a Khovanov-Lauda-Rouquier-Webster algebra. In this way, we obtain a categorical action on category Os for truncated shifted Yangians.

The notion of monoidal categorification of cluster algebras is introduced by Hernandez and Leclerc, and they provided some examples arising from the representation theory of quantum affine algebras. On the other hand, the cluster algebra structure on the unipotent quantum coordinate ring has a monoidal categorification via the representations of symmetric quiver Hecke algebras. Since there is a close connection between the representation theory of quantum affine algebras and that of symmetric quiver Hecke algebras, it is natural to ask whether one can use the results on quiver Hecke algebras to produce monoidal categorifications of cluster algebras in the category of representations over quantum affine algebras. In this talk, I will explain some results along this line of consideration. It is a joint work with S.-j. Kang, M. Kashiwara, E. Park, and S.-j. Oh.

One of the upshots of categorification constructions in representation theory is a nice stock of examples of actions of groups (e.g. braid groups) on triangulated categories. The goal of this talk will be to explain how, following an analogy with the study of mapping class groups of surfaces via Teichmuller theory, such categorical constructions can be used to study the groups themselves.

I will discuss the connections between quantum Heisenberg categories and categorified quantum groups. This is based on joint work with Alistair Savage and Ben Webster.

Coulomb branches have recently been given a good mathematical footing thanks to work of Braverman-Finkelberg-Nakajima. We will discuss their categorical structure. For concreteness we focus on the massless case which leads us to the category of coherent sheaves on the affine Grassmannian (the so called coherent Satake category). This category is conjecturally governed by a cluster algebra structure. We will describe a solution to this conjecture in the case of general linear groups and discuss extensions of this result to more general Coulomb branches of 4D $N=2$ gauge theories. This is joint work with Harold Williams.

We define "q-cut" and "q-flow" lattices associated to a finite graph, arising from a categorification construction for the lattices of integer cuts and flows (which we'll introduce in the talk). We'll describe combinatorial properties of these invariants, including categorified/q-versions of some classical theorems in graph theory. Based on joint work with Tony Licata, available at arxiv:1905.03067.

A quantum symmetric pair a la G. Letzter consists of a quantum group and its coideal subalgebra (called an $\imath$-quantum group). A quantum group can be viewed as an example of $\imath$-quantum groups associated to symmetric pairs of diagonal type, and various fundamental constructions for quantum groups (such as $R$-matrices and canonical bases) have been generalized to $\imath$-quantum groups. In this talk, we present a realization of (universal) $\imath$-quantum groups via modified Ringel-Hall algebras of $\imath$-quivers; this approach leads to PBW bases and braid group actions for $\imath$-quantum groups. Time permitting, we will also discuss a geometric realization of universal $\imath$-quantum groups via Nakajima-Keller-Scherotzke quiver varieties, which produces dual canonical bases with positivity. This is joint work with Ming LU (Sichuan, China).

I will define what is an abelian envelope of a fixed rigid symmetric monoidal category, in the sense of Deligne. I will also give several examples and applications, both in characteristic zero and in positive characteristic.

The category of Soergel bimodules is a well-behaved categorification of the Hecke algebra of a Coxeter group. In many characteristic 0 realizations, the indecomposable objects in this category correspond to the Kazhdan-Lusztig basis, thereby giving an explanation for the positivity of Kazhdan-Lusztig polynomials. In characteristic $p>0$ the indecomposable objects give rise to another set of non-negative Laurent polynomials called $p$-Kazhdan-Lusztig polynomials, which can be used as a replacement for Kazhdan-Lusztig polynomials in modular representation theory. In this talk I will propose a non-negative replacement for inverse Kazhdan-Lusztig polynomials in positive characteristic.

Let $\mathfrak{g}$ be a semisimple Lie algebra with simple roots $I$, and let $\mathcal{C}$ be a category endowed with a categorical $\mathfrak{g}$-action. Recall that Chuang-Rouquier construct, for every $i \in I$, the Rickard complex acting as an autoequivalence of the derived category $D^b(\mathcal{C})$, and Cautis-Kamnitzer show these define an action of the braid group $B_{\mathfrak{g}}$. As part of an ongoing project with Halacheva, Licata, and Losev we show that the positive lift to $B_{\mathfrak{g}}$ of the longest Weyl group element acts as a perverse auto-equivalence of $D^b(\mathcal{C})$. (This generalises a theorem of Chuang-Rouquier who proved it for $\mathfrak{g}=\mathfrak{sl}_2$.) This implies, for instance, that for a minimal categorification this functor is t-exact (up to shift). Perversity also allows us to "crystallise" the braid group action, to obtain a cactus group action on the set of irreducible objects in $\mathcal{C}$. This agrees with the cactus group action arising from the $\mathfrak{g}$-crystal (due to Halacheva-Kamnitzer-Rybnikov-Weekes).