Basics of group representation theoryJuly 10, 2009

Posted by Akhil Mathew in algebra, representation theory.
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Today, I want to talk a bit about group representation theory. Many of us (such as myself) are interested in representation theory in general and will likely talk more about it in the future, so it will be useful to summarize the essential ideas here to refer back. But the basics are well known and have been discussed at length on other blogs (see, e.g. here, which is discussing the subject right now), so I am merely going to summarize these facts without proofs. The interested reader can read these notes for full details. Then, I’ll mention a property to be used later on.

What is a group representation?

Start with a group ${G}$. At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A representation of ${G}$ is essentially an action of ${G}$ on a finite-dimensional complex vector space ${V}$.

Formally, we write:

Definition 1 A representation of the group ${G}$ is a finite-dimensional complex vector space ${V}$ and a group-homomorphism ${G \rightarrow Aut(G)}$. In other words, it is a group homomorphism ${G \rightarrow GL_n(V)}$, where ${n = \dim \ V}$, and ${GL_n}$ is the group of invertible ${n}$-by-${n}$ matrices.

An easy example is just the unit representation, sending each ${g \in G}$ to the identity matrix. (more…)