Editing System of imprimitivity (section)

== Example ==

To motivate the general definitions,  a definition is first formulated, in the case of finite groups and their representations on finite-dimensional [[vector space]]s.

Let ''G'' be a finite group and ''U'' a representation of ''G'' on a finite-dimensional complex vector space ''H''. The action of ''G'' on elements of ''H'' induces an [[Group action (mathematics)|action]] of ''G'' on the vector subspaces ''W'' of ''H'' in this way:

:<math> U_g W = \{ U_g w: w \in W \}. </math>

Let ''X'' be a set of subspaces of ''H'' such that
* the elements of ''X'' are permuted by the action of ''G'' on subspaces and
* ''H'' is the (internal) algebraic [[direct sum of vector spaces|direct sum]] of the elements of ''X'', i.e.,

:<math> H = \bigoplus_{W \in X} W. </math>

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

Two assertions must hold in the definition above:
* the spaces ''W'' for ''W'' ∈ ''X'' must [[linear span|span]] ''H'', and
* the spaces ''W'' ∈ ''X'' must be [[linearly independent]], that is,

:<math> \sum_{W \in X} c_W v_W = 0, \quad v_W \in W \setminus \{ 0 \} </math>

holds only when all the coefficients ''c''<sub>''W''</sub> are zero.

If the action of ''G'' on the elements of ''X'' is [[Group action (mathematics)#Types of actions|transitive]], then we say this is a transitive system of imprimitivity.

Let ''G'' be a finite group and ''G''<sub>0</sub> a subgroup of ''G''. A representation ''U'' of ''G'' is induced from a representation ''V'' of ''G''<sub>0</sub> if and only if there exist the following: 
* a transitive system of imprimitivity (''U'', ''X'') and 
* a subspace ''W''<sub>0</sub> ∈ ''X'' 
such that ''G''<sub>0</sub> is the stabilizer subgroup of ''W'' under the action of ''G'', i.e.

:<math> G_0 = \{g \in G: U_g W_0 \subseteq W_0\}.</math>

and ''V'' is equivalent to the representation of ''G''<sub>0</sub>
on ''W''<sub>0</sub> given by ''U''<sub>''h''</sub> | ''W''<sub>0</sub> for ''h'' ∈ ''G''<sub>0</sub>. 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 show that a [[well-defined]] inducing construction exists on equivalence of representations by considering the [[character theory|character]] of a representation ''U'' defined by

:<math> \chi_U(g) = \operatorname{tr}(U_g). </math>

If a representation ''U'' of ''G'' is induced from a representation ''V'' of ''G''<sub>0</sub>, then

:<math> \chi_U(g) = \frac{1}{|G_0|} \sum_{\{ x \in G : {x}^{-1} \,
g \, x \in G_0\}} \chi_V({x}^{-1} \ g \ x), \quad \forall g \in G. </math>

Thus the character function χ<sub>''U''</sub> (and therefore ''U'' itself) is completely determined by χ<sub>''V''</sub>.

=== Example ===

Let ''G'' be a finite group and consider the space ''H'' of complex-valued functions on ''G''. The left [[regular representation]] of ''G'' on ''H'' is defined by

:<math> [\operatorname{L}_g \psi](h) = \psi(g^{-1} h). </math>

Now ''H'' can be considered as the algebraic direct sum of the one-dimensional spaces ''W''<sub>''x''</sub>, for ''x'' ∈ ''G'', where

:<math> W_x = \{\psi \in H: \psi(g) = 0, \quad \forall g \neq x\}.</math>

The spaces ''W''<sub>''x''</sub> are permuted by L<sub>''g''</sub>.