# Plancherel measure on set of irreducible representations of a finite group

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## Definition

Suppose $G$ is a finite group. Let $\operatorname{Irr}(G)$ be the set of irreducible representations (up to equivalence) of $G$ over the field $\mathbb{C}$ of complex numbers. The Plancherel measure on this set assigns to each element of $\operatorname{Irr}(G)$ the measure $d^2/|G|$ where $d$ is the degree of the representation.

The Plancherel measure is a probability measure in the sense that the total measure of the set is $1$. This follows from the fact that sum of squares of degrees of irreducible representations equals group order.