Representation theory of the symmetric group: basics

In this post we lay out some introductory material on the representation theory of $ S_n$ over $ \mathbb{C}$.

Representation theory: Reducibility and Maschke's theorem

Reductionism being one of our most useful ways of thinking about math, in this post we discuss assumptions under which any representation can be decomposed into a number of `atomic' representations.

Representation theory: Introduction

In this post, we develop the basics of representation theory, loosely following some notes from Dennis Gaitsgory's Math 122 at Harvard. For the most part, we motivate definitions and give different ways of thinking about them; what we say applies to representations of any group over any field.

Fourier analysis on finite abelian groups

This post is an introduction to the Fourier analysis of finite abelian groups, based on notes from Tim Gowers' part III course `Techniques in non-Abelian additive combinatorics', with the occasional bits of hand-wavy intuition.