[Math] Reference request for projective representations of finite groups over a non-problematic field

Question

I would like to get a reference where I can learn about the theory of projective representations of finite groups over the complex numbers (or over any field K such that the order of the given group under study is invertible in K). And how one can relate this to character theory for linear representations (with which I am more familiar). Thanks.

Answer 1

Many textbooks cover this material. For example, Curtis and Reiner, Representation Theory of Finite Groups and Associative Alegbras, Wiley,1962. Projective representations of finite groups (in the sense of Schur) are just genuine linear representations of central extensions. They often arise in Clifford theory, which can be viewed as the decomposition of representations in the presence of normal subgroups. An often encountered situation is to have an irreducible $\mathbb{C}G$-module $V$ whose restriction to a normal subgroup $N$ is a direct sum of isomorphic irreducible $\mathbb{C}N$-modules, say all isomorphic to $U$. In such a situation, there is an action of $G$ by inner automorphisms on ${\rm End}_{\mathbb{C}}(U)$. This almost gives an action of $G$ on $U$ itself, but not quite- the way that an element of $G$ must act on $U$ is only unique up to a scalar multiple. Hence $U$ affords a projective representation of $G$, but not always a genuine linear representation. However, this gives rise to a $2$-cocycle of $G$, and thus to finite central extension of $G$, say ${\hat G}$, and $U$ becomes a genuine $\mathbb{C}{\hat G}$-module. This in turn gives a tensor decomposition of $V$ ( but as a $\mathbb{C}{\hat G}$-module), of the form $V = U \otimes W$, where $W$ is another irreducible $\mathbb{C}{\hat G}$-module.