# Plancherel measure

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In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group $G$ , 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 $G$ being the finite symmetric group $S_{n}$ – see below. It is named after the Swiss mathematician Michel Plancherel for his work in representation theory.

## Definition for finite groups

$\mu (\pi )={\frac {({\mathrm {dim} }\,\pi )^{2}}{|G|}},$ ## Definition on the symmetric group $S_{n}$ An important special case is the case of the finite symmetric group $S_{n}$ , where $n$ is a positive integer. For this group, the set $S_{n}^{\wedge }$ of irreducible representations is in natural bijection with the set of integer partitions of $n$ . For an irreducible representation associated with an integer partition $\lambda$ , its dimension is known to be equal to $f^{\lambda }$ , the number of standard Young tableaux of shape $\lambda$ , so in this case Plancherel measure is often thought of as a measure on the set of integer partitions of given order n, given by

$\mu (\lambda )={\frac {(f^{\lambda })^{2}}{n!}}.$ The fact that those probabilities sum up to 1 follows from the combinatorial identity

$\sum _{\lambda \vdash n}(f^{\lambda })^{2}=n!,$ which corresponds to the bijective nature of the Robinson–Schensted correspondence.

## Application

Plancherel measure appears naturally in combinatorial and probabilistic problems, especially in the study of longest increasing subsequence of a random permutation $\sigma$ . As a result of its importance in that area, in many current research papers the term Plancherel measure almost exclusively refers to the case of the symmetric group $S_{n}$ .

### Connection to longest increasing subsequence

$L(\sigma )=\lambda _{1},\,$ where $\lambda _{1}$ denotes the length of the first row of $\lambda$ . Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of $\lambda$ is exactly the Plancherel measure on $S_{n}$ . So, to understand the behavior of $L(\sigma )$ , it is natural to look at $\lambda _{1}$ with $\lambda$ chosen according to the Plancherel measure in $S_{n}$ , since these two random variables have the same probability distribution. 

### Poissonized Plancherel measure

$\mu _{\theta }(\lambda )=e^{-\theta }{\frac {\theta ^{|\lambda |}(f^{\lambda })^{2}}{(|\lambda |!)^{2}}},$ ### Plancherel growth process

The Plancherel growth process is a random sequence of Young diagrams $\lambda ^{(1)}=(1),~\lambda ^{(2)},~\lambda ^{(3)},~\ldots ,$ such that each $\lambda ^{(n)}$ is a random Young diagram of order $n$ whose probability distribution is the nth Plancherel measure, and each successive $\lambda ^{(n)}$ is obtained from its predecessor $\lambda ^{(n-1)}$ by the addition of a single box, according to the transition probability

$p(\nu ,\lambda )={\mathbb {P} }(\lambda ^{(n)}=\lambda ~|~\lambda ^{(n-1)}=\nu )={\frac {f^{\lambda }}{nf^{\nu }}},$ So, the Plancherel growth process can be viewed as a natural coupling of the different Plancherel measures of all the symmetric groups, or alternatively as a random walk on Young's lattice. It is not difficult to show that the probability distribution of $\lambda ^{(n)}$ in this walk coincides with the Plancherel measure on $S_{n}$ . 

## Compact groups

The Plancherel measure for compact groups is similar to that for finite groups, except that the measure need not be finite. The unitary dual is a discrete set of finite-dimensional representations, and the Plancherel measure of an irreducible finite-dimensional representation is proportional to its dimension.

## Abelian groups

The unitary dual of a locally compact abelian group is another locally compact abelian group, and the Plancherel measure is proportional to the Haar measure of the dual group.

## Semisimple Lie groups

The Plancherel measure for semisimple Lie groups was found by Harish-Chandra. The support is the set of tempered representations, and in particular not all unitary representations need occur in the support.