I lived in Alabama over a summer, and I’m pretty sure I’ve never been as hot and drained from the heat as I’ve been over the last couple of days. So it has been hard to get motivated to post lately. I think I’ll just get a few basics down today, since I’m sort of starting a new topic.
Suppose you have a group and a k-representation . Then the character of G afforded by T, is the map by . So it is the mapping into the field that sends an element to the trace of the linear transformation given by the representation. We call the character of an irreducible representation an irreducible character.
So to do our trivial example that we always have, the “principal character” of G is . Note that the character is not necessarily a homomorphism.
A nifty fact (one that we’d hope is true) is that equivalent representations afford the same character. This is simply because , so , and hence if S is equivalent to T, then which shows they afford the same character.
Another nifty fact is that characters are constant on conjugacy classes of G. This is for essentially the same reason.
To get a few more basics down, a direct sum representation affords the character with the subscripts matching the representation to its character.
I’m not sure how far I’m going to take this. I have a budding idea for a series of posts, but I’m going to do some hunting first to make sure no one else has already done it.