Date Speaker Title Abstract 30 May Adam Block Introduction We will first introduce Summer UMS and then decide on the logistics for the ensuing weeks. 6 June Adam Block Introduction to Representation Theory We will introduce the basics of the theory of representations of finite groups over $\mathbb{C}$ and give some examples. 13 June David Grabovsky The Gelfand-Tsetlin Basis; or, Too Many Direct Sums, and Also a Graph The symmetric groups $S_n$, consisting of all permutations on a set of $n$ elements, naturally contain each other like Matryoshka dolls. ($S_{n-1}$ simply fixes the nth element permuted by $S_n$.) In this talk, we will explore the hope that the representation theory of $S_n$ is also inductive. Along the way, we will develop a tool called the branching graph to help us organize the way that the irreducible representations of $S_n$ decompose into those of lower $S_k$. This decomposition results in the canonical Gelfand-Tsetlin basis for each irreducible representation of $S_n$. Finally, we will construct the Gelfand-Tsetlin algebra and (if time permits) prove that its spectrum uniquely identifies elements from the Gelfand-Tsetlin basis. 20 June Micah Gay Young-Jucys-Murphy Elements In this lecture, we will be considering the branching multigraph of irreducible representations of the $S_n$, although the morals of the arguments are applicable to more general cases. We will liberally apply criteria about the centralizer of a subrepresentation to show that that the restriction of an irreducible representation to a subrepresentation has simple multiplicity, which will show that the branching graph of irreducible representations of $S_n$ is in fact simple. We will then define the Young-Jucys-Murphy elements in $C[S_n]$, show that they in fact generate the Gelfand-Tsetlin algebra, and see how they relate to the GZ-basis. 27 June Ryan Abbott Coxeter Generators and Degenerate Affine Hecke Algebras We will consider the spectra of the YJM elements, defining the set $Spec(n)$ and outlining how we will use this set to construct the representations of $S_n$. We will then use the Coxeter Generators of $S_n$ along with their relations to the YJM elements to motivate the definition of the Degenerate Affine Hecke Algebra $H(2)$. By studying the representations of $H(2)$, we will gain valuable information on the elements of $Spec(n)$, which will bring us closer to constructing the representations of $S_n$. 4 July N/A N/A No meeting due to the holiday. 11 July Myeonhu Kim Content Vectors and the Young Graph; Continuation of Analysis of $Spec(n)$ We continue to pursue our goal to describe the set $Spec(n)$ and the equivalence relation defined on it. First, we will define $Cont(n)$; the set of content vectors of length $n$ ; under the motivation to further restrict possible vectors that can be in $Spec(n)$. We will then define the Young graph and some related notions, and show that there is a bijection between $Cont(n)$ and the set of Young tableaux $Tab(n)$ which also preserves the equivalence class in each set. This discussion will not only enable us to fulfill our goal stated above but also help to obtain an explicit model of representations of $S_n$ and derive the formula for their characters. 18 July Matthew Lerner-Brecher Young Formulas and Miscellaneous Results We’ll begin by building upon the previous lecture to prove the equivalence between $Spec(n)$ and $Cont(n)$. From there, we’ll move on to provide two descriptions of $V^{\lambda}$ by constructing a basis and showing how the Coxeter generators act on said basis. Lastly, we’ll prove some miscellaneous results such as: a bound on the multiplicity of irreducible representations and a description of the centralizer $Z(l,k)$. Time permitting, we might also introduce the Murnaghan–Nakayama rule. 25 July Quang Dao 1 August Ben Church