# 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 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.