Editing Character theory (section)

==Mackey decomposition==
The Mackey decomposition was defined and explored by [[G. Mackey|George Mackey]] in the context of [[Lie group]]s, but is a powerful tool in the character theory and representation theory of finite groups. Its basic form concerns the way a character (or module) induced from a subgroup {{mvar|H}} of a finite group {{mvar|G}} behaves on restriction back to a (possibly different) subgroup {{mvar|K}} of {{mvar|G}}, and makes use of the decomposition of {{mvar|G}} into {{math|(''H'', ''K'')}}-double cosets.

If <math display="inline"> G = \bigcup_{t \in T} HtK </math> is a disjoint union, and {{mvar|θ}} is a complex class function of {{mvar|H}}, then Mackey's formula states that

:<math>\left( \theta^{G}\right)_K = \sum_{ t \in T} \left(\left [\theta^{t} \right ]_{t^{-1}Ht \cap K}\right)^{K},</math>

where {{math|''θ<sup>t</sup>''}} is the class function of {{math|''t''<sup>−1</sup>''Ht''}} defined by {{math|''θ<sup>t</sup>''(''t''<sup>−1</sup>''ht'') {{=}} ''θ''(''h'')}} for all {{mvar|h}} in {{mvar|H}}. There is a similar formula for the restriction of an induced module to a subgroup, which holds for representations over any [[ring (mathematics)|ring]], and has applications in a wide variety of algebraic and [[topology|topological]] contexts.

Mackey decomposition, in conjunction with Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functions {{mvar|θ}} and {{mvar|ψ}} induced from respective subgroups {{mvar|H}} and {{mvar|K}}, whose utility lies in the fact that it only depends on how conjugates of {{mvar|H}} and {{mvar|K}} intersect each other. The formula (with its derivation) is:

:<math>\begin{align}
\left \langle \theta^{G},\psi^{G} \right \rangle &= \left \langle \left(\theta^{G}\right)_{K},\psi \right \rangle \\
&= \sum_{ t \in T} \left \langle \left( \left [\theta^{t} \right ]_{t^{-1}Ht \cap K}\right)^{K}, \psi \right \rangle \\
&= \sum_{t \in T} \left \langle \left(\theta^{t} \right)_{t^{-1}Ht \cap K},\psi_{t^{-1}Ht \cap K} \right \rangle, 
\end{align}</math>

(where {{mvar|T}} is a full set of {{math|(''H'', ''K'')}}-double coset representatives, as before). This formula is often used when {{mvar|θ}} and {{mvar|ψ}} are linear characters, in which case all the inner products appearing in the right hand sum are either {{math|1}} or {{math|0}}, depending on whether or not the linear characters {{math|''θ<sup>t</sup>''}} and {{mvar|ψ}} have the same restriction to {{math|''t''<sup>−1</sup>''Ht'' ∩ ''K''}}. If {{mvar|θ}} and {{mvar|ψ}} are both trivial characters, then the inner product simplifies to {{math|{{pipe}}''T''{{pipe}}}}.