Imprimitive group

A group $ G $ of one-to-one mappings (permutations, cf. Permutation) of a set $ S $ onto itself, for which there exists a partition of $ S $ into a union of disjoint subsets $ S _ {1}, \dots, S _ {m} $, $ m \geq 2 $, with the following properties: the number of elements in at least one of the sets $ S _ {i} $ is greater than $ 1 $; for any permutation $ g \in G $ and any $ i $, $ 1 \leq i \leq m $, there exists a $ j $, $ 1 \leq j \leq m $, such that $ g $ maps $ S _ {i} $ onto $ S _ {j} $.

The collection of subsets $ S _ {1}, \dots, S _ {m} $ is called a system of imprimitivity, while the subsets $ S _ {i} $ themselves are called domains of imprimitivity of the group $ G $. A non-imprimitive group of permutations is called primitive.

An example of an imprimitive group is a non-trivial intransitive group $ G $ of permutations of a set $ S $ (see Transitive group): for a system of imprimitivity one can take the collection of all orbits (domains of transitivity, cf. Orbit) of $ G $ on $ S $. A transitive group of permutations $ G $ of a set $ S $ is primitive if and only if for some element (and hence for all elements) $ y \in S $ the set of permutations of $ G $ leaving $ y $ fixed is a maximal subgroup of $ G $.

The notion of an imprimitive group of permutations has an analogue for groups of linear transformations of vector spaces. Namely, a linear representation $ \rho $ of a group $ G $ is called imprimitive if there exists a decomposition of the space $ V $ of the representation $ \rho $ into a direct sum of proper subspaces $ V _ {1}, \dots, V _ {m} $ with the following property: For any $ g \in G $ and any $ i $, $ 1 \leq i \leq m $, there exists a $ j $, $ 1 \leq j \leq m $, such that

$$ \rho ( g) ( V _ {i} ) = \ V _ {j} . $$

The collection of subsets $ V _ {1}, \dots, V _ {m} $ is called a system of imprimitivity of the representation $ \rho $. If $ V $ does not have a decomposition of the above type, then $ \rho $ is said to be a primitive representation. An imprimitive representation $ \rho $ is called transitive imprimitive if there exists for any pair of subspaces $ V _ {i} $ and $ V _ {j} $ of the system of imprimitivity an element $ g \in G $ such that $ \rho ( g) ( V _ {i} ) = V _ {j} $. The group $ \rho ( G) $ of linear transformations of the space $ V $ and the $ G $-module $ V $ defined by the representation $ \rho $ are also called imprimitive (or primitive) if the representation $ \rho $ is imprimitive (or primitive).

Examples. A representation $ \rho $ of the symmetric group $ S _ {n} $ in the $ n $-dimensional vector space over a field $ k $ that rearranges the elements of a basis $ e _ {1}, \dots, e _ {n} $ is transitive imprimitive, the one-dimensional subspaces $ \{ k e _ {1}, \dots, k e _ {n} \} $ form a system of imprimitivity for $ \rho $. Another example of a transitive imprimitive representation is the regular representation of a finite group $ G $ over a field $ k $; the collection of one-dimensional subspaces $ k g $, where $ g $ runs through $ G $, forms a system of imprimitivity. More generally, any monomial representation of a finite group is imprimitive. The representation of a cyclic group of order $ m \geq 3 $ by rotations of the real plane through angles that are multiples of $ 2 \pi / m $ is primitive.

The notion of an imprimitive representation is closely related to that of an induced representation. Namely, let $ \rho $ be an imprimitive finite-dimensional representation of a finite group $ G $ with system of imprimitivity $ \{ V _ {1}, \dots, V _ {n} \} $. The set $ \{ V _ {1}, \dots, V _ {n} \} $ is partitioned into a union of orbits with respect to the action of $ G $ determined by $ \rho $. Let $ \{ V _ {i _ {1} }, \dots, V _ {i _ {s} } \} $ be a complete set of representatives of the different orbits of this action, let

$$ H _ {m} = \ \{ {g \in G } : {\rho ( g) ( V _ {i _ {m} } ) = V _ {i _ {m} } } \} ,\ m = 1, \dots, s , $$

let $ \phi _ {m} $ be the representation of the group $ H _ {m} $ in $ V _ {i _ {m} } $ defined by the restriction of $ \rho $ to $ H _ {m} $, and let $ \rho _ {m} $ be the representation of $ G $ induced by $ \phi _ {m} $. Then $ \rho $ is equivalent to the direct sum of the representations $ \rho _ {1}, \dots, \rho _ {s} $. Conversely, let $ H _ {1}, \dots, H _ {s} $ be any collection of subgroups of $ G $, let $ \phi _ {m} $ be a representation of $ H _ {m} $ in a finite-dimensional vector space $ W _ {m} $, $ m = 1, \dots, s $, and let $ \rho _ {m} $ be the representation of $ G $ induced by $ \phi _ {m} $. Suppose further that $ \{ g _ {m,j} \} _ {j=} 1 ^ {r _ {m} } $ is a system of representatives of left cosets of $ G $ with respect to $ H _ {m} $. Then the direct sum of the representations $ \rho _ {1}, \dots, \rho _ {s} $ is imprimitive, while $ \rho ( g _ {m,j} ) ( W _ {m} ) $, $ j = 1, \dots, r _ {m} $, $ m = 1, \dots, s $, is a system of imprimitivity (here $ W _ {m} $ is canonically identified with a subspace of $ V $).

References

[1]	M. Hall, "Group theory" , Macmillan (1959)
[2]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)

Comments

A domain of imprimitivity is also called a block.