Young Diagrams and Classical Groups
John C. Baez · arXiv (Cornell University) · 2023
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.