# Graduate Student Representation Theory Seminar

This is website for the grad-student run Graduate Student Representation Theory Seminar at UCR. We will be meeting for the Fall 2020 quarter on Thursdays from 12:30-1:50 via Zoom. If you wish to attend, please contact Jonathan Dugan (jondugan@math.ucr.edu) for the url, or for any other questions or comments concerning the seminar.

GSRTS is an extension of the Lie Theory Seminar run on Tuesdays from 12:30-2:00. The website for that seminar is available at https://sites.google.com/view/petersamuelson/lie-theory-seminar.

I would also like to share the Discussion Meeting on Representation Theory 2020 (DMRT 2020), hosted by the Department of Mathematics at the Indian Institute of Science. Please check their website at https://sites.google.com/view/dmrt2020/home for more information.

### Schedule of Speakers Fall 2020

**October 8, 2020:** Organizational Meeting

**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.

**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 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.

**November 5, 2020:** Alexander Pokorny

**November 12, 2020:** Maranda Smith

**November 19, 2020:** Ethan Kowalenko

**November 26, 2020:** Happy Thanksgiving! No meeting today.

**December 3, 2020:** Speaker TBD

**December 10, 2020:** Speaker TBD