Young Diagrams and Classical Groups

TL;DR

This paper explains how Young diagrams are used to classify representations of symmetric and classical groups, providing insights into their structure and applications in combinatorics and representation theory.

Contribution

It offers a detailed exposition of the role of Young diagrams in classifying representations of various classical groups and the symmetric group, with educational context.

Findings

Young diagrams effectively classify symmetric group representations.

Application of Young diagrams to classical groups like GL(n,C), SL(n,C), U(n), SU(n).

Connections between combinatorics and representation theory elucidated.

Abstract

Young diagrams are ubiquitous in combinatorics and representation theory. Here we explain these diagrams, focusing on how they are used to classify representations of the symmetric groups $S_n$ and various "classical groups": famous groups of matrices such as the general linear group $\mathrm{GL}(n,\mathbb{C})$ consisting of all invertible $n \times n$ complex matrices, the special linear group $\mathrm{SL}(n,\mathbb{C})$ consisting of all $n \times n$ complex matrices with determinant 1, the group $\mathrm{U}(n)$ consisting of all unitary $n \times n$ matrices, and the special unitary group $\mathrm{SU}(n)$ consisting of all unitary $n \times n$ matrices with determinant 1. We also discuss representations of the full linear monoid consisting of all linear transformations of $\mathbb{C}^n$. These notes, based on the column This Week's Finds in Mathematical Physics, are made to accompany a series of lecture videos.