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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.

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For most purposes the complex field can be replaced as you observe by a field whose characteristic is relatively prime to the group order. However, the most definitive results usually require a splitting field for the group, containing enough roots of unity. –  Jim Humphreys May 10 '11 at 16:42
Thanks for making this remark. I suppose the theory gets more complicated when one works over fields that are too small. –  Tommaso Centeleghe May 10 '11 at 21:05

3 Answers 3

up vote 13 down vote accepted

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.

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Thanks! I will take a look to C.R. –  Tommaso Centeleghe May 10 '11 at 14:48
Incidentally you also get the tensor product decomposition on the level of projective $G$-representations (it's not necessary to pass to a central extension). This result from Clifford theory, despite its simple proof, is quite useful. –  fherzig May 11 '11 at 7:07

To supplement Geoff's helpful answer, I'd add a further standard reference: Chapter 11 of the 1976 book Character Theory of Finite Groups by I.M. Isaacs (reprinted in an AMS-Chelsea edition). Here and in the old Curtis-Reiner book (Sections 51-53) you get a lot of specifics about how projective representations arise in finite group theory, along with a treatment of the Schur multiplier (and proof that it is also finite). The theory basically originates with Schur a century ago, whose treatment of symmetric groups and related groups is the subject of a 1992 Oxford monograph by P.N. Hoffman and J.F. Humphreys. In modern usage, Schur's "factor sets" get translated into the language of cohomology.

In the case of finite groups, projective representations arise naturally when you have a group and a quotient group to consider, as Geoff points out. But similar ideas occur naturally in physics, for example in dealing with special orthogonal groups as symmetry groups: here representations come up in a physical context and the notion of "spin" of a particle is best explained by lifting a projective representation to the Spin covering group. Now the groups involved may be infinite, but much of the formalism remains the same.

Similarly, when Steinberg set out in his influential 1963 Nagoya paper "Representations of algebraic groups" to study modular representations of Chevalley groups over finite fields, he began by locating projective representations and then investigated how these might lift to covering groups. Here the point is that the interesting groups may be simple, in which case they often have non-simple covering groups in the background: for example, $SL_2(\mathbb{F}_q) \rightarrow PSL_2(\mathbb{F}_q)$. Eventually Steinberg's study of liftings and central extensions led to much deeper ideas in the case of infinite fields.

In all of this it's important to avoid any confusion with the unrelated homological language of projective modules, even though people often study finite group representations using module language.

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Many thanks for your answer, and for the historical note! –  Tommaso Centeleghe May 10 '11 at 21:08

In addition: Karpilovsky, Gregory. The Schur Multiplier. London Math. Soc. Monographs, 1987.

In this book only $\mathbb{C}$ is considered.

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