Representation Theory of the Symmetric Group

The seminar is now complete! A brief log of what we covered is below, and thanks to Andy Yu, we also have a complete set of notes:

Seminar Notes: Representation Theory of the Symmetric Group

Please email Joshua Wiscons [joshua.wiscons@csus.edu] if you have any questions or notice any errors.

About the Seminar Series

This is a six-part series on the representation theory of the symmetric group. The target audience is undergraduate (and beyond) math-interested folk with some exposure to linear algebra. It is a joint offering of Algebra, Number Theory, and Combinatorics (ANTC) seminar and the Algebra & Logic Lab at Sacramento State.

NSF logo The series is made possible in part by support from the National Science Foundation under Grant No. DMS-1954127. Any opinions, findings, and conclusions or recommendations expressed on this website are those of Joshua Wiscons and do not necessarily reflect the views of the National Science Foundation.

Overview

Topics

The symmetric group: basic structure and properties
Introduction to representations and modules for groups
Young tableaux and the classification of the irreducible $\operatorname{Sym(n)}$-modules
Beyond the classical setting: modules with a dimension

Tentative Schedule

Meeting 1: 3:00–4:00, Friday February 12
Meeting 2: 3:00–4:00, Friday February 26
Meeting 3: 3:00–4:00, Friday March 05
Meeting 4: 3:00–4:00, Friday April 02
Meeting 5: 3:00–4:00, Friday April 30
Meeting 6: 3:00–4:00, Friday May 07

Meeting 1 (February 12)

In our first meeting, we’ll discuss our main goals for the series and explore some background on permutations and the symmetric group. The Zoom link and password for our meeting will be distributed via email. Please email joshua.wiscons@csus.edu if you have questions or would like to be added to the email list.

Topics

Welcome and introductions
Framing and goals
Representing permutations: diagrams and cycles

Preparing for the meeting

There is nothing you need to do to prepare! However, if you want to preview what we’ll discuss, you can follow the links in the list of topics above. But maybe, it’ll be more fun to just watch this video. Thanks for the suggestion BC!

Summary from Meeting 1

First meeting complete! Started off with introductions and learned a bit about each other. It was exciting to see folks in the middle of their undergraduate career, towards the end, current grad students, and current faculty all sharing this space together. After introductions, we formed a community agreement, and briefly discussed our goals for the series.

We spent the remaining time (re)introducing permutations and how to represent them using diagrams and using cycle notation. Great to meet you all—looking forward to next time!

Meeting 2 (February 26)

We’ll pick up where we left off last time. Up next, permutation matrices and how to view $\operatorname{Sym}(n)$ as a group of matrices in a canonical way. Exploring other ways to represent $\operatorname{Sym}(n)$ as a group of matrices is essentially what this series is all about! The Zoom link and password for our meeting will be the same as last time. Please email joshua.wiscons@csus.edu if you have questions or would like to be added to the email list.

Topics

Permutation matrices and the natural representation of $\operatorname{Sym}(n)$
Definition of an $\operatorname{Sym}(n)$-representation
Some questions to drive the rest of our work

Preparing for the meeting

There is nothing to prepare, but please take a look over what we did last time and bring any questions you have.

Summary from Meeting 2

The focus this time was on representing permutations with matrices in a natural way via so-called permutation matrices. We defined what a representation is and showed that our method for associating permutations to matrices was indeed a representation that we called the natural representation of $\operatorname{Sym}(n)$. Our hour concluded by recording some (initial versions) of questions:

Are there representations of $\operatorname{Sym}(n)$ other than the natural one? If so what are they?
Are there representations of $\operatorname{Sym}(n)$ of degree $m$ where $m<n$? If so, how small of an $m$ is possible?

See you all next time!

Meeting 3 (March 05)

This time we’ll connect representations of $\operatorname{Sym}(n)$ to the different (but equivalent) point of view of $\operatorname{Sym}(n)$-modules. Later, we’ll use this new point of view to look for submodules of the natural permutation module (corresponding to the natural representation) and find some additional $\operatorname{Sym}(n)$-modules (hence additional representations).

Topics

$\operatorname{Sym}(n)$-modules and the connection with $\operatorname{Sym}(n)$-representations
The natural $\operatorname{Sym}(n)$-module

Preparing for the meeting

Again, there’s nothing to prepare—just bring any questions you have.

Summary from Meeting 3

We started by seeing a $2$-dimensional representation of $\operatorname{Sym}(3)$. (It looked much different than the $3$-dimensional natural representation, but in fact, they are very tightly connected.) The rest of our meeting was devoted to developing the notion of a $\operatorname{Sym}(n)$-module (which an alternative perspective for representations) and exploring the natural module, which is the module corresponding to the natural representation.

Meeting 4 (April 02)

We’ll keep exploring modules (as a way to explore representations), and this time we’ll look inside the natural $\operatorname{Sym}(n)$-module for submodules. The main goal is to identify the standard $\operatorname{Sym}(n)$-module and learn a bit about it.

Topics

Submodules
The trivial and standard $\operatorname{Sym}(n)$-modules

Preparing for the meeting

As usual, there’s nothing to prepare—just bring questions.

Summary from Meeting 4

The focus of the meeting was on finding more modules of $\operatorname{Sym}(n)$-module by looking at submodules of the natural module. We started with an example to find submodules of the natural module for $\operatorname{Sym}(3)$. This ultimately led to the definition of the trivial module and the standard module. We then found a basis for the standard module (so also determined its dimension). Afterwards, we used this basis to build the corresponding representation, which took us back to an example from long ago. We concluded be recasting our main questions in terms of modules and also slippling in the definition of irreducibility.

Meeting 5 (April 30)

We’ll show that the standard module (usually) is irreducible. We’ll then turn our attention to other actions of the symmetric group, and use them to build more $\operatorname{Sym}(n)$-modules.

Topics

A review of what we’ve done
Irreducibility of the standard module
Action of $\operatorname{Sym}(n)$ on ordered and unordered pairs

Summary from Meeting 5

We spent most of the time proving that the standard module, with scalars from $\mathbb{R}$, is irreducible. We then looked at the action of $\operatorname{Sym}(n)$ on ordered and unordered pairs and used the latter to create a 10-dimensional module for $\operatorname{Sym}(5)$.

Meeting 6 (May 07)

We’ll introduce Young tableaux and use them to build the remaining irreducible $\operatorname{Sym}(n)$-modules.

Topics

Young tableaux and the associated $\operatorname{Sym(n)}$-modules
Classification of the irreducible $\operatorname{Sym(n)}$-modules

Summary from Meeting 6

We discussed tableaux and tabloids and how to construct new $\operatorname{Sym(n)}$-modules using them. This ultimately led to the definition of the Specht modules and the main theorem, which classifies the irreducible $\operatorname{Sym(n)}$-modules over fields of characteristic $0$ (like $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$).

It was nice sharing space and learning math together this semester. Hope you all enjoyed it—I certainly did!!