I know everyone and their brother does a series of posts on basic representation theory, and I said I would try to avoid very overtly repeat posts of other math blogs, but I can’t help it. I don’t know representation theory very well at all, and I feel my time has come to wrestle with the beast. This is one of the main points of this blog, so may as well try.
Goal: Carefully build as slowly as I can all the way to the Artin-Wedderburn Theorem.
Beginning: What is a representation. Well, let G be a finite group and V any finite dimensional vector space over . Then a representation of G is a just a homomorphism . So we can already see some nice uses of this. If we choose a basis, then we get a matrix representation (every element is sent to some invertible matrix, and the group operation is preserved). It would be even nicer if this were an embedding, so that we could actually think of the elements of our group as matrices. We say a 1-1 representation is faithful.
We have an arsenal of examples already, but probably don’t even realize it. The trivial representation is just sending every group element to the identity transformation. What I like to call the “almost trivial representation” (term my own, so don’t use this somewhere and expect people to know what you are talking about), is to embed G in for some large n, which we know is possible. Then under this embedding, a group element is either even or odd. If it is even, send it to the identity transformation. If it is odd, send it to the negative identity transformation. Probably a better way to say this is: a representation of is, .
Let’s define the group ring (aka the group algebra). Let k be a field and G a finite group. Then is the set . In a sense, we have formed a vector space over k with basis G. Our addition is . Our multiplication requires slightly more effort: .
This structure will be of great importance soon, but I don’t want to throw too much out there at once. Remember, we’re going to go slowly. But if you want to think ahead, the first thing of tomorrow’s post will be that a representation equips the vector space V with the structure of a -module. And there was nothing special about there. A k-representation equips V with the structure of a -module.