Wednesdays, 7:30pm; Room 507, Mathematics
Topic: Representation Theory of Symmetric Groups
Source: A New Approach to the Representation Theory of Symmetric Groups by Vershik and Okounkov
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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 GelfandTsetlin 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. ($S_{n1}$ 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 GelfandTsetlin basis for each irreducible representation of $S_n$. Finally, we will construct the GelfandTsetlin algebra and (if time permits) prove that its spectrum uniquely identifies elements from the GelfandTsetlin basis.

20 June 
Micah Gay

YoungJucysMurphy 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 YoungJucysMurphy elements in $C[S_n]$, show that they in fact generate the GelfandTsetlin algebra, and see how they relate to the GZbasis.

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 LernerBrecher

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

