Plancherel measure on set of irreducible representations of a finite group
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Definition
Suppose is a finite group. Let
be the set of irreducible representations (up to equivalence) of
over the field
of complex numbers. The Plancherel measure on this set assigns to each element of
the measure
where
is the degree of the representation.
The Plancherel measure is a probability measure in the sense that the total measure of the set is . This follows from the fact that sum of squares of degrees of irreducible representations equals group order.