# Book for representation theory of finite groups over algebraically closed fields of characteristic $0$?

Since I am writing on my thesis, I need more general literature about representation theory of finite groups over algebraically closed fields of characteristic $$0$$ and not only about the typical case of complex numbers. Does anyone of you have references I could use? Online scripts are also appreciated.

Moreover the most books uses the inner product space theory, so a generalization is not so easy without any references... New techniques are required.

Thank you!

• Curtis-Reiner is not enough? Commented Sep 8, 2020 at 13:48

What ends up happening that the representation theory of a finite group $$G$$ is the same over every algebraically closed field $$K$$ of characteristic $$0$$, and in particular is the same over $$\overline{\mathbb{Q}}$$ as any other such field. A clean way to see this is to use Maschke's theorem, which implies that the group algebra $$\overline{\mathbb{Q}}[G]$$ is semisimple (no inner products necessary). By the Artin-Wedderburn theorem it must therefore be a finite product of matrix rings $$M_{n_i}(\overline{\mathbb{Q}})$$ over $$\overline{\mathbb{Q}}$$, each of which corresponds to an irrep of dimension $$n_i$$ defined over $$\overline{\mathbb{Q}}$$, and then further extending scalars to any algebraically closed field $$K$$ of characteristic $$0$$ produces the same finite product of matrix rings $$M_{n_i}(K)$$ but over $$K$$. So the classification of irreducibles over the two fields coincides; more concretely, every representation defined over $$K$$ is actually defined over $$\overline{\mathbb{Q}}$$, and in fact (with slightly more work) over a finite extension of $$\mathbb{Q}$$ (that is, a number field).
Similarly the orthogonality relations hold with no modifications except that you use $$\chi(g^{-1})$$ instead of $$\overline{\chi(g)}$$ (these are the same over $$\mathbb{C}$$ since $$\chi(g)$$ is a sum of roots of unity).