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 15: 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 29:

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 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.