# UCR Graduate Student Representation Theory Seminar

This is the website for the grad-student run Graduate Student Representation Theory Seminar (GSRTS) at UC Riverside, which meets on Thursdays from 12:30-1:50pm via Zoom. If you wish to attend, please contact Jonathan Dugan (jondugan@math.ucr.edu) for the Zoom link, or for any other questions or comments concerning the seminar.

GSRTS is an extension of the Lie Theory Seminar run on Tuesdays from 12:30-1:50pm. The website for that seminar is available at https://sites.google.com/view/petersamuelson/lie-theory-seminar.

### Past Speakers

#### Spring 2022

**May 19, 2022:** Jonathan Dugan

** Title:** Fusion products and Demazure modules for type

*A*current algebras

**Since the 1980s, an important area of study has been the category of finite-dimensional representations of quantum affine algebras. One way to study these representations is to go from a quantum level to the classical level via a graded limit, which becomes a graded representation of the corresponding current algebra; examples include local Weyl modules and Demazure modules. However, the graded limit of a tensor product of quantum affine algebra modules is not necessarily isomorphic to the tensor product of their graded limits. Instead, the correct notion seems to be that of the fusion product introduced by Feigin and Loktev in 1999, but this product is difficult to work with. Recent work by Biswal, Chari, Shereen, and Wand has found graded characters and generators and relations for the fusion product of a local Weyl module with a level 2 Demazure module for the type**

*Abstract:**A*current algebra under suitable conditions. In this talk, I will share my work to expand on these results by removing these conditions.

**May 12, 2022:** Anthony Muljat

** Title:** On a noetherian property of C-modules

**A key result regarding stability of symmetric group representations states that, given a noetherian ring with unity**

*Abstract:**k*, the category FI is “locally noetherian” over

*k*. This means that every FI-submodule of a finitely generated FI-module over

*k*is also finitely generated. We consider the question: what subcategories of FI are locally noetherian? In this talk we show that, for a slightly more general class of categories C with a suitably restricted subcategory A, local noetherianity of C and A are equivalent. We apply this result to some subcategories of FI and FI

_{BC}that are of interest to the theory of representation stability for finite groups (e.g. the alternating groups

*A*

_{n}and the hyperoctahedral subgroups

*D*

_{n}).

**April 28, 2022:** Raymond Matson

** Title:** From Knot Invariants to Double Affine Hecke Algebras

**As if quantum deformations weren’t hard enough, we are going to construct an even more deformed object called a double affine Hecke algebra. Double affine Hecke algebras were first introduced by Cherednik to solve conjectures about Macdonald polynomials. If you sprinkle some geometry into the mix, the extra deformations DAHAs carry end up being quite useful in cleaning up some equations, smoothing out certain varieties, and they even pop up in geometric Langlands. As representation theorists, a reasonable question for us to ask is what kind of representations can we put on these DAHAs. Using a commonly studied knot invariant in topology, I will construct a particularly defining module for one specific DAHA.**

*Abstract:*#### Winter 2022

**March 3, 2022:** Raymond Matson

** Title:** Quantum Groups and Skein Theory

**Quantum groups are a type of noncommutative algebra with a lot of additional structure that are a central focus of study among many representation theorists due to their applications in physics. As interesting as these structures are, their computations can be quite a pain to work with. Skein theory provides a more playful perspective to understand these so-called quantum groups and their representations. If we have enough time, I will attempt to explain the stated skein theory of the simple bigon, which has a surprising amount of algebraic structure.**

*Abstract:***February 24, 2022:** Jonathan Dugan

** Title:** Macdonald Polynomials and Demazure Modules for the Current Algebra 𝔰𝔩

_{n + 1}[

*t*], Part II

**Last time I spoke, I discussed character formulae and Demazure modules for finite-dimensional representations of the Lie algebra 𝔰𝔩**

*Abstract:*_{n + 1}. Today, we will continue that conversation by talking about graded characters of Demazure modules for the current algebra 𝔰𝔩

_{n + 1}[

*t*]. Just as the characters of the finite-dimensional irreducible modules of 𝔰𝔩

_{n + 1}were schur polynomials

*s*

_{λ}(

*z*), we will find that the graded characters of level one Demazure modules are specialized Macdonald polynomials

*P*

_{λ}(

*z*,

*q*, 0). To study the graded characters of level two Demazure modules, in 2015 Wand introduced the Modules

*M*(

*ν*,

*λ*) which interpolate between level one and level two Demazure modules. In 2020 in a paper published on the arXiv, Chari et. al. showed that for certain “compatible” pairs (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}, the graded characters of these modules

*M*(

*ν*,

*λ*) are certain polynomials

*G*

_{ν, λ}(

*z*,

*q*). In this talk, I will discuss this theory, as well as my work to expand this equality to include all pairs of weights (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}.

**February 17, 2022:** Maranda Smith

** Title:** Graded Characters for level 2 Demazure modules over 𝔰𝔬

_{2n}[

*t*]

**In studying the representation theory of any algebra, it is necessary to understand the highest weight modules for that algebra. This has inspired the research surrounding Demazure modules,**

*Abstract:**D*(ℓ,

*λ*) of affine and quantum affine Lie algebras. To study these modules, we move to a maximal parabolic subalgebra, called the current algebra, and examine these modules in this new context. A first step to understanding these modules is to know what their graded characters are. For simply laced Lie algebras, it was shown by Sanderson and Ion that the graded character of

*D*(1,

*λ*) is the Macdonald Polynomial

*P*

_{λ}(

*z*,

*q*, 0). In 2020, it was shown by Biswal, Chari, Shereen, and Wand that the graded character of

*D*(2,

*λ*) are positive linear combinations of Macdonlad polynomials and give explicit formulations. My research is focused on finding similar results for type-

*D*. This talk is VERY informal, and is meant to walk through my process.

**February 10, 2022:** Anthony Muljat

** Title:** FI

_{W}-modules and stability for Weyl group representations

**“Consistent” sequences {**

*Abstract:**V*

_{n}} of

*S*

_{n}-representations are ubiquitous in geometry and combinatorics. In 2012, Church-Ellenberg-Farb introduced FI-modules to study, among other things, the growth of irreps of

*V*

_{n}as

*n*increases. By doing so, they showed that a sharp constraint on this growth—previously defined by Church-Farb as “uniform representation stability”—is equivalent to a finite generation property of the corresponding FI-module. Building on this framework, Wilson (2014) derived analogous results for representations of the classical Weyl groups. We will discuss this machinery with special attention to types

*A*and

*B*/

*C*, and show some motivating applications.

**February 3, 2022:** Jonathan Dugan

** Title:** Macdonald Polynomials and Demazure Modules for the Current Algebra 𝔰𝔩

_{n + 1}[

*t*]

**The notion of a Demazure module arises naturally from the representation theory of affine and quantum affine Lie algebras. An important family of these are the stable Demazure modules**

*Abstract:**D*(ℓ,

*λ*) which admit an action of the current algebra. To study the structure of these modules, an important question to ask is, “What are their graded characters?” In the simply laced case, it was found through the work of Sanderson and Ion that the specialized Macdonald polynomials

*P*

_{λ}(

*z*,

*q*, 0) are the characters of the level one Demazure module

*D*(1,

*λ*), and in 2012 Naoi showed that for types BCF to find the graded characters for

*D*(1,

*λ*), it suffices to study the relationship between level one and level two Demazure modules for type

*A*. To tackle this problem, in 2015 Wand introduced the Modules

*M*(

*ν*,

*λ*) which interpolate between level one and level two Demazure modules. In 2020 in a paper published on the arXiv, Chari et. al. showed for certain pairs (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}that the graded characters of these modules

*M*(

*ν*,

*λ*) are certain polynomials

*G*

_{ν, λ}(

*z*,

*q*). In this talk, I will discuss this theory, as well as my work to expand this equality to include all pairs of weights (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}.

#### Fall 2021

**November 18, 2021:** Joseph Wagner

** Title:** Local Weyl Modules and

*V*(

*ξ*) Modules

**In this talk, I’ll be presenting some key points from the first two sections of the paper, “Demazure Modules, Fusion Products, and**

*Abstract:**Q*-Systems” by Chari and Venkatesh. Specifically, I’ll be defining the current algebra, the local Weyl module, and two equivalent presentations of

*V*(

*ξ*) modules.

**November 4, 2021:** Elliott Vest

** Title:** An Introduction to the Quantized Universal Enveloping Algebra

**Universal enveloping algebras can be seen as the “largest” associative algebra that contains all representations of a Lie algebra. With such an algebra, one can apply a quantum deformation in order to get a quantized enveloping algebra. There are many properties the quantum enveloping algebra has that are inherited from the original as well as properties that don’t transfer. In this talk, we will focus on both the quantum and regular universal enveloping algebras over 𝔰𝔩**

*Abstract:*_{2}(ℂ). We will note the similarities and differences of such structures which should give rise as to why quantum deformations can be interesting to look into. If time permits, we will then generalize the quantized enveloping algebra over any Lie algebra.

**October 28, 2021:** Raymond Matson

** Title:** An Introduction to Supercharacter Theory

**Classical character theory is a way to condense important information of representations in order to classify groups and algebras up to isomorphism. Although character theory for finite groups is mostly fleshed out, some difficult cases, namely algebra groups, remain. These complicated cases have close relations to nilpotent algebras and quite a bit of Lie theory of these nilpotent algebras. As classical tools don’t seem to work well when computing irreducible characters in these cases, supercharacter theory was introduced and is currently the best we have for classifying these groups. We will discuss how to compute supercharacter theories and if there’s time we will compare classical character tables to supercharacter tables with some nice examples.**

*Abstract:***October 21, 2021:** Maranda Smith

** Title:** An Introduction to the Representation theory of Lie Algebras, Part II

**We will briefly recap where we left off last week, having discussed the ability to classify all finite dimensional simple Lie Algebras over ℂ. In this talk, we will take what we have learned about the adjoint representation and generalize to examine all representations of a simple Lie algebra. This will allows us to classify all finite dimensional irreducible representations over said algebra. Time and will permitting, we will discuss what representations of Lie algebras tells us about the representations of the corresponding Lie groups.**

*Abstract:***October 14, 2021:** Jonathan Dugan

** Title:** An Introduction to the Representation Theory of Lie Algebras

**Lie algebras first arose from as a means to study Lie Groups, but are interesting to study in their own right. In particular, we often look at representations of Lie algebras to see how they act on vector spaces. In this talk, we will introduce the representation theory of Lie algebras, and show how in the classical case, we can use representation theory to classify all finite dimensional simple Lie algebras over ℂ. We will also classify all finite dimensional irreducible representations of the simple Lie algebras, and, if time permitting, briefly discuss how the representation theory of Lie algebras helps us understand the representation theory of Lie groups.**

*Abstract:*#### Spring 2021

**May 27, 2021:** Alex Pokorny

** Title:** Dubrovnik Skein Theory and Power Sum Elements

**In this work, we extend some results from the Kauffman bracket and HOMFLYPT skein theories to the Kuffman (Dubrovnik) skein theory. In particular, a definition is given for special “power sum” type elements**

*Abstract:**P̃*

_{k}in the Dubrovnik skein algebra of the annulus 𝒜. It is justified that these elements are the correct generalization for the Chebyshev polynomials (of the first kind) often used when studying Kauffman bracket skein algebras. Threadings of the

*P̃*

_{k}are used as generators in a presentation of the Dubrovnik skein algebra of the torus 𝒯

^{2}, where they are shown to satisfy surprisingly simple relations. This description of 𝒯

^{2}is in turn used to describe the natural action of this algebra on the skein module of the solid torus. Separate from this, we give strong evidence that the universal character rings for the orthogonal and sympletic Lie groups correspond to the skein algebra 𝒜 in such a way that the Schur functions of type either B, C or D correspond to annular closures

*Q̃*

_{λ}of minimal idempotents of the Birman-Murakami-Wenzl algebras

*B*

*M*

*W*

_{n}. We also record some miscellaneous applications of the

*P̃*

_{k}, such as commutation relations for the annular closures of BMW symmetrizers

*Q̃*

_{(n)}(and their HOMFLYPT counterparts

*Q*

_{(n)}) and an expression of central elements of

*B*

*M*

*W*

_{n}in terms of Jucys-Murphy elements.

**May 6, 2021:** Maranda Smith

** Title:** Short Exact Sequences and Polynomial Recursion

**We will briefly review the structure of type**

*Abstract:**D*current algebras and introduce the modules

*M*(

*ν*,

*λ*). As of now there are 5 right exact sequences involving these modules which are conjectured to be full exact. Based on this conjecture, we will discuss what recursions are possible for their graded characters and what this means for the representation theory of highest weight modules over type

*D*current algebras as a whole.

**April 22, 2021:** Jonathan Dugan

** Title:** Macdonald Polynomials and Demazure Modules for the Current Algebra 𝔰𝔩

_{n + 1}[

*t*]: A Detailed Look at Explicit Examples

**In previous talks I have talked about the 𝔰𝔩**

*Abstract:*_{n + 1}[

*t*]-modules

*M*(

*ν*,

*λ*) first introduced by Wand in 2015 which interpolate between level 1 and level 2 Demazure modules. In 2021 Chari et al showed for certain pairs (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}that the graded characters of these modules

*M*(

*ν*,

*λ*) are certain polynomials

*G*

_{ν, λ}(

*z*,

*q*). My research is to expand this equality to include all pairs of weights (

*ν*,

*λ*). In this talk, I will go into the details of how I am approaching this by looking explicitly at the rank 1–3 cases.

#### Winter 2021

**March 11, 2021:** Maranda Smith

** Title:** Beginnings on graded characters of level-2 Demazure modules for type-

*D*

**In 2015 Jefferey Wand introduced a family of modules**

*Abstract:**M*(

*ν*,

*λ*) over type-

*A*current algebras. These modules interpolate between level-1 and level-2 Demazure modules, giving way for Biswal, Chari, Shereen, and Wand to define ch

_{gr}

*M*(

*ν*,

*λ*) as integer linear combination of Macdonald Polynomials. As these modules can be similarly defined for type-

*D*current algebras, the goal is to extend these results to this setting.

**March 4, 2021:** Alex Pokorny

** Title:** Skein Theories Coming From Representation Theory

**A skein theory is an attempt to build a sort of algebraic topology based on links in 3-manifolds. They were invented to generalize the idea of knot polynomials to different manifolds. In this talk I will give some examples of skein theories which emulate and extend parts of the representation theory of certain quantum groups.**

*Abstract:***February 18, 2021:** Joseph Wagner

** Title:** A Brief Introduction to Universal Enveloping Algebras

**I’ll start by motivating the topic, and then talk about how they’re constructed, proving that a representation of 𝔤 is the same thing as a representation of**

*Abstract:**U*(𝔤), and then, time permitting, discuss their bases and the PBW theorem.

**January 28, 2021:** Jonathan Dugan

** Title:** Macdonald Polynomials and Demazure Modules for the Current Algebra 𝔰𝔩

_{n + 1}[

*t*], Part II

**In this talk, I will continue with the story that we began the previous week. I will finish the discussion of Demazure modules by exploring on the work of Chari et al to find their characters using the representation theory of the modules**

*Abstract:**M*(

*ν*,

*λ*). I will then finish by discussing my work to expand on their efforts, along with showing a few examples.

**January 21, 2021:** Jonathan Dugan

** Title:** Macdonald Polynomials and Demazure Modules for the Current Algebra 𝔰𝔩

_{n + 1}[

*t*], Part I

**The notion of a Demazure module arises naturally from the representation theory of affine and quantum affine Lie algebras. An important family of these are the stable Demazure modules**

*Abstract:**D*(ℓ,

*λ*) which admit an action of the current algebra. To study the structure of these modules, an important question to ask is, “What are their graded characters?” In the simply laced case, it was found through the work of Sanderson and Ion that the specialized Macdonald polynomials

*P*

_{λ}(

*z*,

*q*, 0) are the characters of the level one Demazure module

*D*(1,

*λ*), and in 2012 Naoi showed that for types BCF to find the graded characters for

*D*(1,

*λ*), it suffices to study the relationship between level one and level two Demazure modules for type

*A*. To tackle this problem, in 2015 Wand introduced the Modules

*M*(

*ν*,

*λ*) which interpolate between level one and level two Demazure modules. In 2020 in a paper published on the arXiv, Chari et. al showed for certain pairs (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}that the graded characters of these modules

*M*(

*ν*,

*λ*) are certain polynomials

*G*

_{ν, λ}(

*z*,

*q*). In this talk, I will discuss my work to expand this equality to include all pairs of weights (

*ν*,

*λ*) ∈

*P*

^{+}×

*P*

^{+}.

#### Fall 2020

**December 10, 2020:** Alexander Pokorny

** Title:** The Character Rings of Classical Lie Groups & Symmetric Functions

**It’s hard to avoid combinatorics when studying representation theory. In this talk, I will explain one reason for why this statement is true. First we will discuss characters of representations of Lie groups, focusing on the cases of**

*Abstract:**G*

*L*

_{n},

*S*

*O*

_{n}, and

*S*

*P*

_{n}. Then I will explain what the ring of symmetric functions is and how it can model the behavior of these character rings. Later, I will describe the connections to skein theory.

**November 19, 2020:** Ethan Kowalenko

** Title:** Representation Theory for Oriented Matroids

**I’ll talk about oriented matroid programming, which is a generalization of linear programming, and discuss noncommutative algebras which can be defined for “generic” oriented matroid programs. I will then talk about the representation theory of these algebras is a nice subcase. This is joint work with my advisor, Carl Mautner. This will be a very rough version of my defense, so I’ll love to have as much feedback as possible!**

*Abstract:***November 12, 2020:** Maranda Smith

** Title:** Casual Dip into Lie Theory

**We’ll be taking a pretty comfortable and intuitive look at what a Lie algebra is and how to work with their representations. There will be several examples as we go, primarily coming from Types**

*Abstract:**A*and

*D*. We’ll wrap with a brief look at the representations that I specifically work with.

The aim of this talk is to give those of you in Lie Algebras part A some intuition to move forward with. Questions encouraged!

Notes for this talk available here.

**October 29, 2020:** Joseph Wagner

** Title:** Partitions and Tableaux

**In this talk, I will be covering Appendix A2 of H. Barcelo’s and A. Ram’s paper**

*Abstract:**Combinatorial Representation Theory*. I will define some concepts relating to partitions and tableaux, culminating in the theorem known as the Murnaghan-Nakayama rule.

**October 22, 2020:** Jonathan Dugan

** Title:** The Symmetric Group, Representation Theory, and Paradoxes in Voting Theory

**In this talk, I will explain how we can use representation theory of the symmetric group to model voting theory. A voter’s ranking of candidates can be represented using a combinatorial object called a tabloid, and the rankings of a group of voters can be seen as a representation of the group ring ℚ**

*Abstract:**S*

_{n}. We will then turn our attention to positional voting procedures, which assign points to each candidate based on their position in a voter’s choice of tabloid. Lastly, I will present a result of Daugherty et al. that uses representation theory to show that the results of an election can more accurately reflect the choice of voting procedure rather than the views of the voters.

**October 15, 2020:** Alexander Pokorny

** Title:** Hecke Algebras of Coxeter Systems

**In this talk, I will define Coxeter groups and give a summary of their classification. Using this data, we can define so-called Hecke Algebras. We will mostly focus on the Hecke algebra of type**

*Abstract:**A*

_{n}, and its relationship to representations of the quantum group

*U*

_{q}(𝔰𝔩

_{n}). If time permits, I would also very much like to discuss the Brauer and BMW algebras. I will assume very few prerequisites for this talk, so feel free to sit in even if you are unfamiliar. I will take ideas from Appendix B of the following expository paper by Barcelo & Ram https://arxiv.org/pdf/math/9707221.pdf, but also some from my personal notes.