Clifford theory

(for group representations)

Let $ N $ be a normal subgroup of a finite group $ G $ and let $ RG $ be the group algebra of $ G $ over a commutative ring $ R $. Given an $ RN $- module $ V $ and $ g \in G $, let $ ^ {g} V $ be the $ RN $- module whose underlying $ R $- module is $ V $ and on which $ N $ acts according to the rule $ n * v = ( g ^ {- 1 } ng ) v $, $ v \in V $, where $ n * v $ denotes the module operation in $ ^ {g} V $ and $ nv $ the operation in $ V $. By definition, the inertia group $ H $ of $ V $ is $ H = \{ {g \in G } : {V \cong ^ {g} V } \} $. It is clear that $ H $ is a subgroup of $ G $ containing $ N $; if $ H = G $, it is customary to say that $ V $ is $ G $- invariant

Important information concerning simple and indecomposable $ RG $- modules can be obtained by applying (perhaps repeatedly) three basic operations:

i) restriction to $ RN $;

ii) extension from $ RN $; and

iii) induction from $ RN $. This is the content of the so-called Clifford theory, which was originally developed by A.H. Clifford (see [a1]) for the classical case where $ R $ is a field. General references for this area are [a2], [a3].

The most important results are as follows.

Restriction to normal subgroups of representations.

Given a subgroup $ H $ of $ G $ and an $ RG $- module $ U $, let $ U _ {H} $ denote the restriction of $ U $ to $ RH $. If $ V $ is an $ RH $- module, then $ V ^ {G} $ denotes the induced module. For any integer $ e \geq 1 $, let $ eV $ be the direct sum of $ e $ copies of a given module $ V $. A classical Clifford theorem, originally proved for the case where $ R $ is a field, holds for an arbitrary commutative ring $ R $ and asserts the following. Assume that $ U $ is a simple $ RG $- module. Then there exists a simple submodule $ V $ of $ U _ {N} $; for any such $ V $ and the inertia group $ H $ of $ V $, the following properties hold.

a) $ U _ {N} \cong e ( \oplus _ {t \in T } ^ {t} V ) $, where $ T $ is a left transversal for $ H $ in $ G $. Moreover, the modules $ ^ {t} V $, $ t \in T $, are pairwise non-isomorphic simple $ RN $- modules.

b) The sum $ W $ of all submodules of $ U _ {N} $ isomorphic to $ V $ is a simple $ RH $- module such that $ W _ {N} \cong eV $ and $ U \cong W ^ {G} $.

The above result holds in the more general case where $ {G / N } $ is a finite group. However, if $ {G / N } $ is infinite, then Clifford's theorem is no longer true (see [a3]).

Induction from normal subgroups of representations.

The principal result concerning induction is the Green indecomposable theorem, described below. Assume that $ R $ is a complete local ring and a principal ideal domain (cf. also Principal ideal ring). An integral domain $ S $ containing $ R $ is called an extension, of $ R $, written $ {S / R } $, if the following conditions hold:

A) $ S $ is a principal ideal domain and a local ring;

B) $ S $ is $ R $- free;

C) $ J ( S ) ^ {e} = J ( R ) S $ for some integer $ e \geq 1 $. One says that $ {S / R } $ is finite if $ S $ is a finitely generated $ R $- module. An $ RG $- module $ V $ is said to be absolutely indecomposable if for every finite extension $ {S / R } $, $ S \otimes _ {R} V $ is an indecomposable $ SG $- module.

Assume that the field $ {R / {J ( R ) } } $ is of prime characteristic $ p $( cf. also Characteristic of a field) and that $ {G / N } $ is a $ p $- group. If $ V $ is a finitely generated absolutely indecomposable $ RN $- module, then the induced module $ V ^ {G} $ is absolutely indecomposable. Green's original statement pertained to the case where $ R $ is a field. A proof in full generality is contained in [a3].

Extension from normal subgroups of representations.

The best result to date (1996) is Isaacs theorem, described below. Let $ N $ be a normal Hall subgroup of a finite group $ G $, let $ R $ be an arbitrary commutative ring and let $ V $ be a simple $ G $- invariant $ RN $- module. Then $ V $ extends to an $ RG $- module, i.e. $ V \cong U _ {N} $ for some $ RG $- module $ U $. Originally, R. Isaacs proved only the special case where $ R $ is a field. A proof in full generality can be found in [a3].

References

[a1]	A.H. Clifford, "Representations induced in an invariant subgroup" Ann. of Math. (2) , 38 pp. 533–550
[a2]	G. Karpilovsky, "Clifford theory for group representations" , North-Holland (1989)
[a3]	G. Karpilovsky, "Group representations" , 3 , North-Holland (1994)