Plancherel measure

In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group ${\displaystyle G}$, that describes how the regular representation breaks up into irreducible unitary representations. In some cases the term Plancherel measure is applied specifically in the context of the group ${\displaystyle G}$ being the finite symmetric group ${\displaystyle S_{n}}$ – see below. It is named after the Swiss mathematician Michel Plancherel for his work in representation theory.

Let ${\displaystyle G}$ be a finite group, we denote the set of its irreducible representations by ${\displaystyle G^{\wedge }}$. The corresponding Plancherel measure over the set ${\displaystyle G^{\wedge }}$ is defined by

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