What is the background in algebraic geometry and representation theory needed for geometric complexity theory?

Question

What are the topics that should be mastered by someone who wants to understand geometric computational complexity?

Surely, a lot of algebraic geometry and representation theory are needed, but which topics? and representation theory of what? groups, lie algebras etc? naming good resources ( texts etc ) that cover this background will be highly appreciated.

I'm not sure if this question should be asked here in math.se or in theoretical computer science forum, so if you find it more useful to post it there, please tell me.

Answer 1

I took a course with Mulmuley on GCT. I think the main topics to be familiar with are

• representation theory of the symmetric group
• representation theory of $\mathrm{GL}_n \mathbb{C}$
• geometric invariant theory

Fulton and Harris is good for the rep theory (or google around for notes). I guess Mumford's book is the standard for geometric invariant theory (GIT) though frankly I don't know much about it. The basic setting of GIT is algebro-geometric so some knowledge of the basic objects of algebraic geometry would be required.