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)
How to Cite This Entry:
Clifford theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_theory&oldid=46360
This article was adapted from an original article by G. Karpilovsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article