Representation theory (at least the origin of this terminology) aims to exhibit a model (a represetative) in the group of matrices for an abstract group which is known by only its group law. So complex representations is a satisfying beautiful theory with Schur's theorem, Frobenius reciprocity, Artin, Brauer theorems and various results about irreducible degrees for finite groups.


But my concern is that the field of complex numbers is extraneous to groups. Do we have theorems like $\sum\_j d\_j^2 = |G|$ when $d\_j$ runs though all irreducible degrees of $G$ over an unspecified algebraically closed field. Because the complex case uses character theory with Hermitian structure on the vector space of complex-valued class functions I am not sure if this result is available for a general field.


I have not studied what is known as modular representation theory. Are all non-modular cases no different from the theory over complex numbers? (always working over algebraically closed field). Do the degrees of irreducible 
representations divide the order of the group whatever the base field is? 


Instead of asking for questions individually of this nature let me ask:


*Is there an analogue of Lefschetz principle of Algebraic Geometry that is valid for finite group representations?* so that to study non-modular representations it suffices to study complex representations.



Is there any Lefschetz-like principle for representations of finite groups?
I was encouraged to make my comment an answer, so will do so.


If $G$ is a finite group and $\mathbb{K}$ is a field then many interesting results that can be proved using character theory can also be proved by analysing the structure of the group algebra $\mathbb{K}G$. For instance by (27.20) of Curtis and Reiner's "Representation theory of finite groups and associative algebras" we have


$$\mathbb{K}G \cong \bigoplus\_i M\_{d\_i}(\mathbb{K})$$


is a direct sum of matrix algebras whenever $\mathbb{K}$ is an algebraically closed field whose characteristic does not divide the order of the group. Counting dimensions this gives the desired numerical identity


$$|G| = \sum\_i d\_i^2.$$


Note however that dropping either of these conditions can be problematic. It is well known that if the characteristic of $\mathbb{K}$ divides the order of the group then $\mathbb{K}G$ is no longer semisimple. However, if the field is too small then the irreducible representations of $G$ may not be realisable over that field. For instance, take $\mathbb{K} = \mathbb{Q}$ and $G$ to be a cyclic group.


