Clifford's Theorem with all its aspects in modern language, looking for a textbook

Question

I am looking for a (more or less) introductory textbook on representation theory that contains the full contents of Clifford's paper "Representations Induced In An Invariant Subgroup" in modern language. That is to say: The study of how irreducible representations of a group $G$ decompose under restriction to a normal subgroup $N\trianglelefteq G$ of finite index $(G:N)=m$.

Most books I have seen either only contain special cases, such as $m=2$ or $G$ finite, or they are too general, not treating the special case where $G/N$ is cyclic of order $m$, which particularly interests me.

Thanks a lot in advance!

Answer 1

If you are speaking of a finite dimensional irreducible $G$-module (for a possibly infinite group $G$), then there is little difference from the standard argument for finite groups when dealing with a normal subgroup $N$ of finite index. Take an irreducible $N$-submodule $U$. There are only finitely many different (necessarily irreducible) $N$-submodules of the form $Ug$ (with $g \in G$), and these are permuted by $G$ under right translation. The module $V$ (regarded as $N$-module) is a direct sum of some of them, and the elements $g \in G$ such that $Ug \cong U$ form a subgroup $I$ containing $N.$ There are $[G:I]$ non-isomorphic irreducible summands of $V$ viewed as $N$-module, and each of these isomorphism types of irreducible summands occurs with equal multiplicity.

In the important case that $I = G$, and when the field is algebraically closed, the question of extendibility of $U$ to a $G$ module is determined by $2$-cocycles of the finite group $G/N$. In particular, in that case, if $G/N$ is cyclic, then $U$ does extend to a $G$-module.

Answer 2

Probably the most recent textbook which treats this material in a "modern" way is Methods of Representation Theory I (Wiley Interscience, 1981) by Curtis and Reiner. Combined with its volume II, this treatise covers much more ground than the 1962 pioneering text they wrote; it also uses more modern language. Naturally their coverage of Clifford theory comes fairly early in the book (section 11), in the context of induced representations.

Keep in mind that there are relatively few textbook treatments that go this far beyond classical character theory of finite groups. They also include quite a lot of material in modular representation theory as well as algebraic K-theory, etc.

P.S. It's true that Curtis and Reiner normally confine themselves to finite groups, but their treatment can sometimes be adapted to more general groups when dealing just with the algebraic aspects of induction.

Answer 3

Maybe the following book will be useful for you:

G.Karpilovsky. Group Representations, Vol. 1, Part B: Introduction to Group Representations and Characters. Elsevier, 1992.

See in it Sec.18.11. Clifford’s theorem and further.

In particular, Lemma 3.2 in Sec.23.3. begins with the words:

"Let $N$ be a normal subgroup of $G$ such that $G/N$ is cyclic of order $n$."

Addendum: E.C.Dade, Clifford theory and induction from subgroups. In: Representation theory, group rings, and coding theory, Pap. in Honor of S. D. Berman, Contemp. Math. 93, 133-144 (1989).