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Basics of group representation theory
*July 10, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: algebra, groups, linear algebra, representation theory

6 comments

Tags: algebra, groups, linear algebra, representation theory

6 comments

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

Formally, we write:

Definition 1A representation of the group is a finite-dimensional complex vector space and a group-homomorphism . In other words, it is a group homomorphism , where , and is the group of invertible -by- matrices.

An easy example is just the unit representation, sending each to the identity matrix. (more…)