# Lie Theory Seminar

This is the (minimal) website for the Lie Theory Seminar at UCR for the Fall 2022 quarter. Schedules for various previous quarters are lower down on the page.

The graduate students organize a Lie Theory Seminar for graduate students, as an extension of the Lie Theory Seminar to give talks on different areas in Representation Theory. The website is here.

## Speakers F2022

This quarter talks will start at 12:30 + epsilon because I'm teaching till 12:20.

September 27: Organizational Meeting

October 4: Raymond Matson, UCR

October 11: (no speaker)

October 18: (no speaker)

October 25: Vijay Higgins, MSU

November 1:

November 8: Catherine Cannizzo, UCR

November 15: Mark Ebert, USC

November 22: No talk -- Thanksgiving

November 29:

## Titles and Abstracts: F2022

**September 27**: Organizational Meeting **October 4**: Stated Skein Modules and DAHAs, Raymond Matson

Abstract: Some knot invariants come from looking at highly noncommutative associated groups. As these groups can be incredibly difficult to work with, one can instead consider corresponding commutative algebras and representations. However, if you want to still extract knot invariants you need to quantize these algebras and skein theory provides a breathable way to understand these deformations. In 2012, Berest and Samuelson provided a geometric way to understand these invariants and in the process uncovered certain defining modules for a bigger underlying beast, double affine Hecke algebras. I will discuss these module structures and how they act in the context of a newer, more general skein theory recently established by Thang Lê. This new theory, called stated skein theory, provides significant additional algebraic structure to these algebras and modules and will hopefully lead to more insights into a nicer presentation of these DAHAs. **October 11**:**October 18**:**October 25**: Stated skein algebras for Kuperberg webs, Vijay Higgins

A skein algebra is spanned by links in a thickened surface subject to skein relations. Le introduced a finer skein algebra with extra skein relations along the boundary of the surface, called the stated skein algebra. Stated skein algebras are compatible with cutting and gluing of surfaces. When the skein relations are the ones associated to SL(2), Costantino and Le showed that the stated skein algebra encodes the quantum group OqSL(2).

In this talk we will explore the situation when the skein relations are Kuperberg's web relations for SL(3) or for Sp(4). We will focus on constructing bases for the stated skein algebras and relating these algebras to quantum groups and their representation categories.**November ****1**:**November ****8**: Global homological mirror symmetry for genus 2 curves, Catherine Cannizzo

We will describe the categorical correspondence in homological mirror symmetry between a complex genus 2 curve and its symplectic generalized SYZ mirror. One main idea of the proof is that a 4-torus is SYZ mirror to a 4-torus. We will motivate this with the example of T², the elliptic curve, mirror to a symplectic 2-torus. The global result for all genus 2 curves, namely allowing the complex and symplectic structures to vary in their real six-dimensional families, is joint work with H. Azam, H. Lee, and C-C. M. Liu.

**November 1****5**: Derived Superequivalences for Spin Symmetric Groups and Odd sl2-categorifications, Mark Ebert, USC

Since Chuang and Rouquier's pioneering work showing that categorical sl(2)-actions give rise to derived equivalences, the construction of derived equivalences has been one of the more prominent tools coming from higher representation theory. In this talk, we explain joint work with Aaron Lauda and Laurent Vera giving new super analogues of these derived equivalences stemming from the odd categorification of sl(2). Just as Chuang and Rouquier used their equivalences to prove Broué's abelian defect conjecture for symmetric groups, we use our superequivalences to prove this long standing conjecture for spin symmetric (and spin alternating) groups. **November 2****9**:

## Speakers F2021

October 12: Organizational Meeting

October 19: An introduction to the sl(2) double affine Hecke algebra, Peter Samuelson

October 26: Knot invariants from Hopf algebras, Matthew Harper

November 2:

November 9: Alexandra Utiralova

November 16: Nate Harman

November 23: Rank 2 Jones-Wenzl projectors, Elijah Bodish

November 30: Hongdi Huang

## Titles and Abstracts: F2021

**October 12: **Organizational meeting**October 19:** We will give a gentle introduction to the sl(2) DAHA. We will define the algebra and discuss its origin and some basic properties. We will also define the polynomial representation and some basic properties of Macdonald polynomials. **October 26: ***Knot invariants from Hopf algebras***November 2: **No talk this week.**November 9:** *Harish-Chandra bimodules in complex rank.*

Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogues of classical objects provide insights on their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in Deligne categories that is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of Deligne categories.**November 16: **Admissible Representations of Infinite-Rank Arithmetic Groups

I'm going to talk about a surprisingly well-behaved class of representations of infinite-rank arithmetic groups such as SL_\infty(Z). Before this though I will explain the better known analogous theory for the infinite symmetric group S_\infty, as well as give a crash course in the finite dimensional representation theory of arithmetic groups like SL_n(Z).

**November 23**: In 1932 Rumor-Teller-Weyl observed that endomorphism algebras of tensor products of the vector representation for sl_2 can be described by (linear combinations of) crossingless matching diagrams. This is now well-known under the moniker “Temperley-Lieb algebra”. The Jones-Wenzl projectors are linear combinations of crossingless matching diagrams which describe the idempotent projecting to the symmetric powers. These projectors satisfy recursive formulas which can aid in their computation, which have proved useful in representation theory, knot theory, the study of subfactors, and categorification.

In his 1996 paper “Rank 2 spiders for Lie algebras", Kuperberg defined an analogue of Tempeley-Lieb algebras for each rank 2 simple Lie algebra (i.e. sl_3, sp_4, and g_2). He also proved that the analogues of the Jones-Wenzl projectors exist for rank 2 as well. Then D. Kim and later B. Elias found recursive descriptions of these projectors in the case of sl_3, and Elias gave a conjecture about how these recursions may look for sl_n. The most interesting aspect of this conjecture is that the coefficients in the projectors, which are by definition solutions to some complicated recursive formula, are actually described compactly by formulas analogous to the Weyl dimension formula.

In my talk I will review the above background material and then discuss my work on finding recursive descriptions of the projectors in the case of sp_4 and g_2 (sp_4 appears in arxiv:2102.05186 and g_2 is joint work in progress with Haihan Wu from UC Davis). I will also discuss how the rank 2 projectors fit into the framework of Elias’s conjecture in type A, and suggest how the whole story may generalize to other Lie algebras. **November 30**: Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation

Let H be a Hopf algebra over a field k such that H is Z-graded as an algebra. In this talk, we introduce the notion of a twisting pair for H and show that the Zhang twist of H by such a pair can be realized as a 2-cocycle twist. As an application of twisting pairs, we discuss an algorithm to produce a family of solutions to the quantum Yang-Baxter equation from a given solution via the Faddeev-Reshetikhin-Takhtajan construction.

## Speakers S2021

March 30: Eugene Gorsky, Cyclotomic expansions for gl(N) knot invariants via interpolation Macdonald polynomials

April 6: no talk

April 13: no talk

April 20: no talk

April 27: Brian Collier, Positivity and nilpotents with applications to character varieties

May 4: Oded Yacobi, Categorical representations of braid groups and Kazhdan-Lusztig bases

May 11: no talk

May 18: no talk

May 25: no talk

June 1: Carl Mautner, Symmetric products of the plane, symmetric groups and perverse sheaves

## Titles and Abstracts: S2021

**March 30: **Eugene Gorsky, Cyclotomic expansions for gl(N) knot invariants via interpolation Macdonald polynomials

We construct a new basis for the cyclotomic completion of the center of the quantum gl(N) in terms of the interpolation Macdonald polynomials. This allows us to define a cyclotomic expansion of the universal gl(N) knot invariants generalizing those of Habiro for sl(2). This is a joint work with Anna Beliakova.

**April 27**: Brian Collier, Positivity and nilpotents with applications to character varieties

In this talk we will discuss recent work of Guichard-Wienhard on positive structures on flag varieties of real reductive Lie groups. Positivity leads to the notion of positive surface group representations and conjecturally a description of all “higher Teichmüller spaces”. For split groups, positivity is Lusztig positivity and positive surface group representations are known as Hitchin representations. We will then give an alternate description/classification of positive structures in terms of special conditions on sl(2) triples in the complexified Lie algebra. Hopefully we will have time to hint at how this alternate description leads to a parameterization of components of the character variety via Higgs bundles.

**May 4**: Oded Yacobi, Categorical representations of braid groups and Kazhdan-Lusztig bases

We'll explain some new (and old) results about the action of the symmetric group on the Kazhdan-Lusztig basis of a Specht module. Our tools come from the categorical representations of braid groups, and more specifically those representations which arise as Rickard complexes acting on the bounded derived category of an sl_n categorification. We will assume no prior knowledge about categorical representation theory.

**June 1**: Carl Mautner, Symmetric products of the plane, symmetric groups and perverse sheaves

I will discuss work in progress with Tom Braden in which we study categories of modular perverse sheaves on symmetric products of the plane. Using the geometry of the Hilbert scheme of points, we relate these categories to the symmetric group and its representation ring.

## Speakers W2021

January 12: Jacob Greenstein, Quantum Grothendieck resolutions

January 19: Jacob Greenstein

January 26: Deniz Kus, Prime representations in the HL category

February 2: Ryo Fujita, Positivity of an analog of Kazhdan-Lusztig polynomials for finite-dimensional representations of quantum affine algebras

February 9: no seminar

February 16: no seminar

February 23: Jianrong Li, Grassmannian cluster algebras

March 2: Tamanna Chatterjee, Parity sheaves arising from graded Lie algebras

March 9: Bogdan, Ion, Stable Daha’s and the double Dyck path algebra

March 16: final exams

## Titles and Abstracts: W2021

**January 12: **Quantum Grothendieck resolutions, Jacob Greenstein

I will discuss classical, quantum and representation theoretic aspects of Grothendieck resolutions of a semisimple Lie group and its Poisson dual group. The link between the algebraic (respectively, quantum) versions of these resolutions is provided by the algebraic (respectively, quantum version) of the celebrated Semenov-Tian-Shansky map.

**January 26: **Prime representations in the HL category, Deniz Kus

Abstract: Generators and relations of graded limits of certain finite dimensional irreducible representations of quantum affine algebras have been determined in recent years. For example, the representations in the Hernandez-Leclerc category corresponding to cluster variables appear to be certain truncations of representations for current algebras and tensor products are related to the notion of fusion products. In this talk we will discuss some known results on this topic and study the classical characters of prime representations in the HL category.

**February 2**: Positivity of an analog of Kazhdan-Lusztig polynomials for finite-dimensional representations of quantum affine algebra, Ryo Fujita

Abstract: For a complex simple Lie algebra $\mathfrak{g}$, finite-dimensional representations of the associated untwisted quantum affine algebra form an interesting monoidal abelian category, which has been studied from various perspectives. Related to the fundamental problem of determining the characters of irreducible representations in this category, one can consider an analog of Kazhdan-Lusztig polynomials in a purely algebraic way. When $\mathfrak{g}$ is of simply-laced type, the positivity of these polynomials follows from Nakajima's geometric theory of quiver varieties, which is not applicable to non-simply-laced cases. In this talk, we show that the same positivity holds for non-simply-laced type as well by establishing an isomorphism between the quantum Grothendieck ring of non-simply-laced type and that of ''unfolded'' simply-laced type. In addition, we newly find that an analog of Kazhdan-Lusztig conjecture holds for several cases in non-simply-laced type. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.

**February 23**: Grassmannian cluster algebras, Jianrong Li

Abstract: In this talk, I will talk about Grassmannian cluster algebras and their connection with representations of quantum affine algebras, representations of p-adic groups, and Grassmannian cluster categories. This is based on join work with Wen Chang, Bing Duan, Chris Fraser, and with Karin Baur, Dusko Bogdanic, Ana Garcia Elsener.

**March 2**: Parity sheaves arising from graded Lie algebras, Tamanna Chatterjee

Abstract: Let G be a complex, connected, reductive, algebraic group, and $\chi: C^* \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $\chi(C^*)$. Here I will talk about $G_0$-equivariant parity sheaves on the n-graded piece, $\mathfrak{g}_n$. For the first half we will spend on derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. I will define parabolic induction and restriction in graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumption together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on $\mathbb{Z}$-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

**March 9**: Stable Daha’s and the double Dyck path algebra. Bogdan, Ion

Abstract: The double Dyck path algebra (ddpa) is the algebraic structure that governs the phenomena behind the shuffle and rational shuffle conjectures. It was introduced by Carlsson and Mellit as the key character in their proof of the shuffle conjecture and later Mellit used it to give a proof of the rational shuffle conjecture. While the structure emerged from their considerations and computational experiments while attacking the conjecture, it bears some resemblance to the structure of a double affine Hecke algebra (daha) of type A. Carlsson and Mellit mentioned the clarification of the precise relationship as an open problem. I will explain how the entire structure and its standard (defining) representation emerges naturally and canonically from a stable limit of the family of $GL_n$ daha’s. From this perspective a new commutative family of operators comes forth. Their spectral properties are still to be explored. This is joint work with Dongyu Wu

## Speakers F2020

October 6: Wee Liang Gan, UCR. Notes

October 13: Wee Liang Gan, UCR. Notes

October 20: Pablo Ocal, Texas A&M.

October 27: Peter Samuelson, UCR.

November 3: Peter Samuelson, UCR.

November 10: Elie Casbi, Paris 7.

November 17: Alistair Savage, University of Ottowa.

November 24:

December 1: Lea Bitmann, University of Vienna.

December 8:

## Titles and Abstracts: F2020

**October 6:** Wee Liang Gan FI-modules: Proof of Representation Stability

Abstract: I will explain the notion of representation stability and the proof that finitely generated FI-modules are representation stable.

**October 13**: Wee Liang Gan FI-modules: Proof of Noetherianity

Abstract: I will explain the proof that finitely generated FI-modules over a noetherian ring are noetherian.

**October 20**: Pablo Ocal, Hochschild cohomology of general twisted tensor products

Abstract: The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. This structure is useful because of its applications in deformation and representation theory, and recently in quantum symmetries. Unfortunately, computing it remains a notoriously difficult task. In this talk we will present techniques that give explicit formulas of the Gerstenhaber algebra structure for general twisted tensor product algebras. This will include an unpretentious introduction to this cohomology and to our objects of interest, as well as the unexpected generality of the techniques. This is joint work with Tekin Karadag, Dustin McPhate, Tolulope Oke, and Sarah Witherspoon.

**October 27**: Peter Samuelson, Skein relations in representation theory and geometry, I

Abstract: In this survey talk I discuss different ways that the SL(2) skein relations appear in representation theory and geometry, including quantum groups, double affine Hecke algebras, and character varieties.

**November 3**: Peter Samuelson, Skein relations in representation theory and geometry, I

Abstract: In this continuation I discuss different ways that the GL(infinity) skein relations appear in representation theory and geometry, including quantum groups, the Goldman Lie algebra, and Hall algebras. If time permits I will discuss what is known in the SO(infinity) case also.

**November 10**: Elie Casbi, Equivariant multiplicities of simply-laced type flag minors

Abstract: The study of remarkable bases of (quantum) coordinate rings has been an intensive research area since the early 90’s. For instance the multiplicative properties of these bases (in particular the dual canonical basis) was one of the main motivations for the introduction of cluster algebras by Fomin and Zelevinsky around 2000. In a recent work, Baumann-Kamnitzer-Knutson introduced an algebra morphism D from the coordinate ring of a maximal unipotent subgroup N to the function field of a maximal torus T. It is related to the geometry of Mirkovic-Vilonen cycles via the notion of equivariant multiplicity. This morphism turns out to be useful to compare good bases of C[N]. We will focus on comparing the values taken by D on several distinguished elements of the Mirkovic-Vilonen basis and the dual canonical basis. For the latter one, we will use Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of C[N] via quiver Hecke algebras as well as recent results by Kashiwara-Kim. This will lead us to an explicit description of the images under D of the flag minors of C[N] as well as remarkable identities between them.

**November 17:** Alistair Savage, Affinization of monoidal categories

Abstract: We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.

**December 1:** Lea Bittmann, Quantum Grothendieck rings, towards simply laced types

Abstract: The quantum Grothendieck rings of certain categories of representations of quantum affine algebras admit remarkable t-deformations called quantum Grothendieck rings. This notion was first introduced, via the study of quiver varieties, for categories of finite-dimensional representations, when the underlying Lie algebra is of simply laced type. This construction was then algebraically generalized to non simply laced types. In this talk, we will focus on a category O^+ of representations, and we will see how the quantum Grothendieck ring of this category can be define combinatorally, as a quantum cluster algebra. In particular, we will discuss this construction in non simply laced types.

## Speakers W2020

January 23: open

January 30: Andrew Manion, USC.

February 6: open

February 13: Ethan Kowalenko

February 20: open

February 27: Justin Davis, UCR.

March 5: Matthew Burns

March 12: cancelled ~~Yilong Wang~~

Spring Quarter: At this point, we expect no talks will be held this quarter due to COVID-19.

April 2:

April 9: ~~Siddharth Venkatesh~~

## Titles and Abstracts: W2020

**January 30:** Andrew Manion, Heegaard Floer algebras, hypertoric varieties, and the amplituhedron (joint with A. Lauda and A. Licata)

Recently, Ozsvath-Szabo showed that their 2016 theory of "bordered knot Floer homology" does indeed compute knot Floer homology. This theory has a rich algebraic structure with many relationships to other areas of mathematics. I will sketch the physical framework into which Heegaard Floer homology is supposed to fit, along with the expected role of Ozsvath-Szabo's bordered knot Floer homology in the overall framework. Then I will discuss a new observation that apparently goes beyond the existing physical framework, namely that Ozsvath-Szabo's algebras from bordered knot Floer homology can be viewed as convolution algebras for certain hypertoric varieties whose associated polytopes are relatives of the "amplituhedron" as introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in maximally supersymmetric gauge theory.

**February 13**: Ethan Kowalenko, Representation Theory associated to Oriented Matroid Programs

Abstract: Around 2008, Braden, Licata, Proudfoot, and Webster associated a finite-dimensional algebra to any sufficiently generic linear programming problem, viewed as a hyperplane arrangement. The representation category of this algebra is highest weight, and the algebra is Koszul. These statements are also true of the algebra determined by the principal block of BGG category O from Lie theory. In this talk, I will describe these linear programs via their combinatorics and define the associated algebra, pointing out the poset that gives rise to the highest weight structure of the representation category. I will state a few theorems that were proved by Braden-Licata-Proudfoot-Webster, and then describe what additional assumptions we need to make in order to obtain these theorems in a generalization to oriented matroid programs. Time permitting, I'll describe some questions we have yet to answer. The generalization to oriented matroids is joint work with Carl Mautner.

**February 27**: Justin Davis, Generalized Demazure Modules with Level Two Demazure Filtrations

Abstract**:** This work is motivated by graded limits of a family of irreducible prime representations of the quantum affine algebra associated to a simply-laced simple Lie algebra g introduced by David Hernandez and Bernard Leclerc in the context of monoidal categorification of cluster algebras. The graded limit of a member of this family is an indecomposable graded module for the current algebra g[t]. In a recent paper with V. Chari and R. Moruzzi Jr., we studied the Type D_n case. We showed that in certain cases, the limit is a so-called generalized Demazure module. We gave a presentation for this module, and computed its graded character in terms of level two Demazure modules.

**March 5: **Matthew Burns, A survey on Lie groups

Abstract: This talk will serve as an introduction to the theory of Lie groups. We will discuss the relationship between Lie algebras and Lie groups via the exponential map and explore their cohomology groups. Many examples will be presented. A lot of information about the Lie algebra can be encoded in these groups and in certain cases can actually yield information about the cohomology of the corresponding Lie group.

**March 12**: Yilong Wang, Higher central charge and the Witt group of modular tensor categories

Abstract: Modular tensor categories (MTCs) are generalizations of finite abelian groups equipped with non-degenerate quadratic forms. Via the associated topological quantum field theory, they are closedly related to topology and physics. They also have deep connections with arithmetics through the action of the absolute Galois group on the category.

This talk contains three parts. In the first part, we will give a brief introduction to MTCs and their Witt group, generalizing the classical Witt group of non-degenerate quadratic forms. The second part of the talk will focus on the arithmetic properties of numerical invariants of MTCs called the higher central charges. Finally, we will show an application of higher central charges to the study of the Witt group.