### Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

## Talks

Thursday, July 15, 12:00 ~ 12:35 UTC-3

## Parabolic Hilbert schemes and rational Cherednik algebras

### Monica Vazirani

#### UC Davis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

One can use the geometry of parabolic Hilbert schemes of points on plane curve singularities to build representations of the rational Cherednik algebra (RCA) in type A. We use an alternate presentation of the RCA to describe these geometric representations. The basis of equivariant homology that comes from torus fixed points has a nice combinatorial description; and furthermore, this basis diagonalizes the action of the Dunkl-Opdam subalgebra of the RCA. We make use of our explicit combinatorial bases as well as the alternate presentation to construct explicit maps between standard modules parameterized by hooks, thus recovering the BGG resolution of the simple module parameterized by the trivial hook.

Joint work with Eugene Gorsky (UC Davis) and José Simental (Max-Planck-Institut für Mathematik).

Thursday, July 15, 12:45 ~ 13:20 UTC-3

## Subspace arrangements and submodules of the polynomial representation of the rational Cherednik algebra

### Stephen Griffeth

#### Universidad de Talca, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Representation theory may sometimes be used to study the ideals of varieties with large symmetry groups: for instance, this happens with determinantal varieties. I will discuss the extent to which the representation theory of Cherednik algebras may be applied in this fashion to the ideals of arrangements of linear subspaces, and advertise some related conjectures on the submodule structure of the polynomial representation of the Cherednik algebra of a cyclotomic reflection group.

Thursday, July 15, 13:30 ~ 14:05 UTC-3

## Restriction of square integrable representations

### Jorge Vargas

#### Famaf-CIEM, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be a semisimple Lie group, and $(\pi,V)$ a irreducible square integrable representation for $G$. Thus, a model for $V$ is the $L^2$-kernel of a elliptic operator on a fiber bundle over the symmetric space $G/K$ attached to $G$. Let $H$ be a closed reductive subgroup for $G$. We say $\pi$ is $H$-discretely decomposable ( $H$-admissible) if the sum of the closed $H$-irreducible subspaces in $V$ is dense in $V$, ($H$-admissible if it is $H$-discretely decomposable and the multiplicity of each irreducible factor is finite). We give criteria for being $H$-$\cdots$ in language of spherical functions as well as in the language of differential intertwining operators. On a basic exposition we will present an overview of some aspects of branching problems and results in Orsted-Vargas, Branching problems in reproducing kernel spaces, Duke mathematical journal, Vol. 169, 3478-3537, 2020 and some consequences.

Thursday, July 15, 14:35 ~ 15:10 UTC-3

## Theta operators, Macdonald polynomials, and new symmetric function operator identities

### Marino Romero

#### University of California, San Diego, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Theta operators are symmetric function operators that were first introduced by D'Adderio, Iraci and Wyngaerd in order to give the Delta Conjecture a compositional refinement. It was then also conjectured that the Frobenius characteristic of the space of coinvariants in two sets of commuting and two sets of anticommuting variables can be given using the Theta operators (extending a conjecture of Zabrocki). We will present several new symmetric function operator identities that can be specialized to give Theta operator identities. In turn, many identities in the literature regarding Delta eigenoperators of the modified Macdonald basis become consequences of our identities. We will also discuss two of the main tools used in proving our identities: Tesler's Identity and Garsia-Mellit's Five Term Relation, two fundamental identities in the theory of modified Macdonald polynomials. Part of our presentation is based off joint work with Michele D'Adderio.

Joint work with Michele D'Adderio (Université Libre de Bruxelles).

Thursday, July 15, 15:20 ~ 15:55 UTC-3

## Uncrowding algorithm for hook-valued tableaux

### Anne Schilling

#### University of California Davis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm ``uncrowds'' the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that ``uncrowds'' the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.

Joint work with Jianping Pan (UC Davis, USA), Joseph Pappe (UC Davis, USA) and Wencin Poh (UC Davis, USA).

Friday, July 16, 16:00 ~ 16:35 UTC-3

## Light leaves and its avatars

### Nicolas Libedinsky

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Light leaves are some bases of the morphism spaces between Soergel bimodules. After their apparition, several other incarnations of the same idea have emerged (or are emerging): anti-spherical, indecomposable, canonical, and singular light leaves. I will explain all of these avatars.

Joint work with Ben Elias (University of Oregon), Hankyung Ko (Uppsala University), Leonardo Patimo (University of Freiburg) and Geordie Williamson (University of Sydney).

Friday, July 16, 16:45 ~ 17:20 UTC-3

## Quantum geometric Satake and K-theory

### Ben Elias

#### University of Oregon, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We explain a new (conjectural) quantum version of the geometric Satake equivalence, using K-theoretic Soergel bimodules. We explain the connection with an earlier quantum geometric Satake equivalence, which uses a q-deformation of ordinary Soergel bimodules.

Joint work with Geordie Williamson (U. Sydney).

Friday, July 16, 17:30 ~ 18:05 UTC-3

## Mixed perverse sheaves on flag varieties of Coxeter groups

### Cristian Vay

#### UNC & CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We construct an abelian category of ``mixed perverse sheaves'' attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of Achar and Riche, where they worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

Joint work with Pramod N. Achar (Louisiana State University, USA) and Simon Riche (Université Clermont Auvergne, France).

Friday, July 16, 18:35 ~ 19:10 UTC-3

## Combinatorial invariance conjecture for $\widetilde{A}_2$

### David Plaza

#### Universidad de Talca, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig polynomials are equal, i.e., $P_{x,y}(q)=P_{x',y'}(q)$. In this talk we prove this conjecture for the affine Weyl group of type $\widetilde{A}_2$. This is the first non-trivial case where this conjecture is verified for an infinite group.

Joint work with Gastón Burrul (The University of Sidney, Australia), and Nicolás Libedinsky (Universidad de Chile, Chile).

Friday, July 16, 19:20 ~ 19:55 UTC-3

## Costandard Whittaker modules and contravariant pairings

### Anna Romanov

#### University of Sydney, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.

In 1997, Milicic—Soergel introduced a category N of modules over a semisimple Lie algebra which includes both category O and all Whittaker modules. In many ways, the structure of this category is similar to category O: objects are finite-length, simple objects arise as unique irreducible quotients of parabolically-induced standard modules, and composition multiplicities are given by Kazhdan—Lusztig polynomials. However, in other ways, the category is surprising: the objects are not weight modules, and standard modules do not admit unique contravariant forms. In particular, the lack of weight-space decompositions means that the duality in category O cannot be naively extended to category N. In ongoing work with Brown, we classify contravariant pairings between standard Whittaker modules and Verma modules, which leads to a natural algebraic definition of costandard objects in category N. We show that these costandard objects align with costandard (twisted) Harish-Chandra sheaves under Beilinson—Bernstein localization, and that with this set of costandard objects, category N has the structure of a highest weight category.

Joint work with Adam Brown (Institute of Science and Technology, Austria).

Tuesday, July 20, 16:00 ~ 16:35 UTC-3

## Braid varieties, weaves, and positroids.

### José Simental Rodríguez

#### Max Planck Institute for Mathematics, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

To a positive braid we associate an affine algebraic variety that we call the braid variety. These varieties are closely related to and in some cases generalize well-known varieties that appear in Lie theory such as Richardson, positroid and open Bott-Samelson varieties. They are also closely related to the augmentation variety of a Legendrian link associated to the corresponding positive braid. I will define the braid varieties and explain some of their properties and the connections mentioned above as well as a diagrammatic calculus, the weaves from the title, to study them that is closely related to Soergel calculus but differs from it in key aspects. Time permitting, I will also explore the question on how to define these varieties for braid words which are not necessarily positive, and explain consequences for positroid varieties.

Joint work with Roger Casals (University of California, Davis), Eugene Gorsky (University of California, Davis) and Mikhail Gorsky (Université de Picardie Jules Verne).

Tuesday, July 20, 16:45 ~ 17:20 UTC-3

## Tautological classes and symmetry in Khovanov-Rozansky homology

### Eugene Gorsky

#### University of California, Davis, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We define a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that $F_2$ satisfies "hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen.

Joint work with Matt Hogancamp (Northeastern University, USA) and Anton Mellit (University of Vienna, Austria).

Tuesday, July 20, 17:30 ~ 18:05 UTC-3

## Link homologies and Hilbert schemes via representation theory

### Tina Kanstrup

#### University of Massachusetts Amherst, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The aim of this joint work in progress with Roman Bezrukavnikov is to unite different approaches to Khovanov-Rozansky triply graded link homology. The original definition is completely algebraic in terms of Soergel bimodules. It has been conjectured by Gorsky, Negut and Rasmussen that it can also be calculated geometrically in terms of cohomolgy of sheaves on Hilbert schemes. Motivated by string theory Oblomkov and Rozansky constructed a link invariant in terms of matrix factorizations on related spaces and later proved that it coincides with Khovanov-Rozansky homology. In this talk I'll discuss a direct relation between the different constructions and how one might invent these spaces starting directly from definitions.

Tuesday, July 20, 18:50 ~ 19:25 UTC-3

## Upsilon-like invariants from Khovanov homology

### Melissa Zhang

#### University of Georgia, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will survey link concordance invariants coming from Khovanov homology, particularly those similar in spirit to Ozsváth-Stipsicz-Szabó's Upsilon, a 1-parameter family of invariants coming from knot Floer homology. This is related to my joint work with Linh Truong on annular link concordance invariants as well as ongoing work with Ross Akhmechet.

Joint work with Linh Truong (University of Michigan, USA) and Ross Akhmechet (University of Virginia, USA).

Tuesday, July 20, 19:35 ~ 20:10 UTC-3

## Generators and relations for $\text{Rep}(Sp_{2n})$

### Elijah Bodish

#### University of Oregon, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\text{Rep}(G)$ be the monoidal category of finite dimensional representations of a semisimple Lie group. Kuperberg’s 1996 paper “Spiders for rank 2 Lie algebras” proposed studying $\text{Rep}(G)$ by first studying $\text{Fund}(G)$, the full monoidal subcategory generated by finite dimensional irreducible modules with highest weight a fundamental weight. Kuperberg went on to give generators and relations for $\text{Fund}(G)$ when $G = SL_3$, $Spin_5$, $Sp_4$, and $G_2$. The problem of giving analogous generators and relations for$\text{Fund}(SL_n)$ was solved in 2012 by Cautis—Kamnitzer—Morrison.

In joint work with Elias, Rose, and Tatham (https://arxiv.org/abs/2103.14997) we define a category by generators and relations and argue there is functor from the generators and relations category to $\text{Fund}(Sp_{2n})$. Combining skein theoretic arguments with combinatorial results due to Sundaram, as well as exploiting a well known relation between BMW algebras and symplectic groups, we deduce the functor is an equivalence of monoidal categories, solving Kuperberg’s problem for $\text{Fund}(Sp_{2n})$.

In the talk I will begin with an expanded discussion of the history outlined above. Then state our results and try to give an idea of how some of the arguments work by illustrating them in the case of $Sp_4$ and $Sp_6$.

Joint work with Ben Elias (University of Oregon), David Rose (UNC-Chapel Hill) and Logan Tatham (UNC-Chapel Hill).

Tuesday, July 20, 20:20 ~ 20:55 UTC-3

## The Kauffman/BMW skein algebra of the torus

### Peter Samuelson

#### University of California, Riverside, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The BMW algebra surjects onto centralizer algebras of tensor powers of the defining representation of the type BCD quantum groups (like the Hecke algebra in type A). Kauffman found that skein relations defining this algebra can be used to define invariants of knots in $\mathbb{R}^3$. These skein relations associate algebras to surfaces, and we give a presentation of the algebra associated to the torus. At the end we ask whether there is a $q,t$ deformation of this algebra which can be used to describe BCD knot homology of (iterated) torus knots, similar to the Hall algebra of elliptic curves defined by Burban and Schiffmann.

Joint work with Hugh Morton (University of Liverpool) and Alex Pokorny (University of California, Riverside).