System of Imprimitivity - Illustrative Example

Illustrative Example

To motivate the general definitions, we first formulate a definition in the case of finite groups and representations of these on finite dimensional vector spaces.

Suppose G is a finite group and U is a representation of G on a finite-dimensional complex vector space H. The action of G on elements of H induces an action of G on a vector subspace W of H in an obvious way:

Suppose X is a set of subspaces of H such that (1) the elements of X are permuted by the action of G on subspaces and (2) H is the (internal) algebraic direct sum of the elements of X, viz.

Then (U, X) is a system of imprimitivity for G.

Two assertions must hold in the above definition:

1. the spaces W for W ∈ X must span H and
2. the spaces W ∈ X must be linearly independent, that is the linear relation

only holds when all the coefficients c_W are zero.

If the action of G on the elements of X is transitive, then we say this is a transitive system of imprimitivity.

Suppose G is a finite group, G₀ a subgroup of G. A representation U of G is induced from a representation V of G₀ if and only if there exist the following:

• a transitive system of imprimitivity (U, X) and
• a subspace W₀ ∈ X

such that G₀ is the fixed point subgroup of W under the action of G, i.e.

and V is equivalent to the representation of G₀ on W₀ given by U_h | W₀ for h ∈ G₀. Note that by this definition, induced by is a relation between representations. We would like to show that there is actually a mapping on representations which corresponds to this relation.

For finite groups one can easily show that a well-defined inducing construction exists on equivalence of representations by considering the character of a representation U defined by

In fact if a representation U of G is induced from a representation V of G₀, then

Thus the character function χ_U (and therefore U itself) is completely determined by χ_V.