Clifford theory

(for group representations)

Let be a normal subgroup of a finite group and let be the group algebra of over a commutative ring . Given an -module and , let be the -module whose underlying -module is and on which acts according to the rule , , where denotes the module operation in and the operation in . By definition, the inertia group of is . It is clear that is a subgroup of containing ; if , it is customary to say that is -invariant

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

i) restriction to ;

ii) extension from ; and

iii) induction from . 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 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 of and an -module , let denote the restriction of to . If is an -module, then denotes the induced module. For any integer , let be the direct sum of copies of a given module . A classical Clifford theorem, originally proved for the case where is a field, holds for an arbitrary commutative ring and asserts the following. Assume that is a simple -module. Then there exists a simple submodule of ; for any such and the inertia group of , the following properties hold.

a) , where is a left transversal for in . Moreover, the modules , , are pairwise non-isomorphic simple -modules.

b) The sum of all submodules of isomorphic to is a simple -module such that and .

The above result holds in the more general case where is a finite group. However, if 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 is a complete local ring and a principal ideal domain (cf. also Principal ideal ring). An integral domain containing is called an extension, of , written , if the following conditions hold:

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

B) is -free;

C) for some integer . One says that is finite if is a finitely generated -module. An -module is said to be absolutely indecomposable if for every finite extension , is an indecomposable -module.

Assume that the field is of prime characteristic (cf. also Characteristic of a field) and that is a -group. If is a finitely generated absolutely indecomposable -module, then the induced module is absolutely indecomposable. Green's original statement pertained to the case where 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 be a normal Hall subgroup of a finite group , let be an arbitrary commutative ring and let be a simple -invariant -module. Then extends to an -module, i.e. for some -module . Originally, R. Isaacs proved only the special case where 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)