Dimensions of irreducible representations of finite groups over $\mathbb Q$ - Mathematics Stack Exchange If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the cyclic group linear representation theory page degrees of irreducible representations over splitting field degrees of irreducible representations over reals degrees of irreducible representations over rationals smallest splitting field (characteristic zero) smallest size of finite splitting field 1 : trivial group-- 1 : 1 : 1 : 2, i.e., field:F2: 2 Math. 55 (1990), no.
Computing irreducible representations of finite groups. Let L be an irreducible representation, then, there is a surjective map A → L from the regular representation (just map 1 to a non-zero element). A = k G the group algebra of a finite group which is the case you are asking about).
Babai and Ronyai.
If the group is given by its multiplication table then there are polynomial-time algorithms for all the tasks you mentioned (over $\mathbb{C}$; these don't handle modular representations). Comp. In particular, L is finite dimensional. Let A be a finite dimensional algebra (e.g. 192, 705–722.