GARTS Fall 2023

Sept. 15, 2023

Speaker: Madison Crim

Title: Introduction to Representation Theory of Lie Algebras

Abstract: I will provide background on what a Lie algebra is including some examples. We’ll discuss semisimple Lie algebras which give rise to important results of finite dimensional representations such as Weyl’s theorem on complete reducibility. These tools will then allow us to examine representations of the simple Lie algebra sl(2, F).

Sept. 22, 2023

Speaker: Ching Hsien Lee

Title: Root Systems

Abstract: The classification of simple Lie algebras arises from their correspondence with root systems. The root systems consist of finite sets of vectors in Euclidean space, possessing distinctive symmetry and integral characteristics. We will delve into this connection and elucidate how it impacts the structure of simple Lie algebras and their associated modules.

Sept. 29, 2023

Speaker: Sumit Singh

Title: Freudenthal's Multiplicity formulas

Abstract: The Problem addressed in this talk is about finding the dimension of Weight Spaces of Irreducible representation of finite dimensional Lie Algebra corresponding to a dominant weight. This can be solved recursively using Freudenthal Multiplicity Formula. The main idea in the proof is to calculate the action of Universal Casimir Element, which acts as a scalar on the space.

Oct. 6, 2023

Speaker: Dan Tan

Title: PBW Theorem

Abstract: Universal enveloping algebras relate the representation theory of Lie algebras to the representation theory of associative unital algebras. The PBW Theorem provides a basis for a given universal enveloping algebra. Continuing our series of introductory talks on Lie algebra, we’ll state and prove the PBW Theorem as presented in Humphreys.

Oct. 27, 2023

Speaker: Dennis Hou

Title: The Infinite Symmetric Group

Nov. 3 & 10, 2023

Speaker: Hong Chen

Title: Symmetric Functions and Characters of Symmetric Groups

Abstract: I will first give an introduction to the algebra of symmetric functions, including three sets of free generators $e_n$, $h_n$ and $p_n$ (elementary, complete homogeneous and power sum), two bases $m_\lambda$ and $s_\lambda$ (monomial and Schur), many relations among them, and a scalar product. Then I will talk about the relation between symmetric functions and characters of symmetric groups, in particular, I will show that the algebra of symmetric functions is isometrically isomorphic to the algebra of characters of symmetric groups. I will also mention the hook-length formula.

Dec. 1, 2023

Speaker: Nick Backes

Title: TBD

Abstract: TBD

Spring 2021 Talks (Jason Saied)

Spring 2019 Talks (Alejandro Ginory)