# Plancherel measure

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

Let $G$ be a finite group, we denote the set of its irreducible representations by $G^{\wedge }$. The corresponding Plancherel measure over the set $G^{\wedge }$ is defined by
    $\mu (\pi )={\frac {(\mathrm {dim} \,\pi )^{2}}{|G|}},$
where $\pi \in G^{\wedge }$, and $\mathrm {dim} \pi$ denotes the dimension of the irreducible representation $\pi$.

## Definition on the symmetric group

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

Let $L(\sigma )$ denote the length of a longest increasing subsequence of a random permutation $\sigma$ in $S_{n}$ chosen according to the uniform distribution. Let $\lambda$ denote the shape of the corresponding Young tableaux related to $\sigma$ by the Robinson–Schensted correspondence. Then the following identity holds:
    $L(\sigma )=\lambda _{1},$
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

Plancherel measure is defined on $S_{n}$ for each integer $n$. In various studies of the asymptotic behavior of $L(\sigma )$ as $n\rightarrow \infty$, it has proved useful to extend the measure to a measure, called the Poissonized Plancherel measure, on the set ${\mathcal {P}}^{*}$ of all integer partitions. For any $\theta >0$, the Poissonized Plancherel measure with parameter $\theta$ on the set ${\mathcal {P}}^{*}$ is defined by
    $\mu _{\theta }(\lambda )=e^{-\theta }{\frac {\theta ^{|\lambda |}(f^{\lambda })^{2}}{(|\lambda |!)^{2}}},$
for all $\lambda \in {\mathcal {P}}^{*}$.

### 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
    $p(\nu ,\lambda )=\mathbb {P} (\lambda ^{(n)}=\lambda ~|~\lambda ^{(n-1)}=\nu )={\frac {f^{\lambda }}{nf^{\nu }}},$
for any given Young diagrams $\nu$ and $\lambda$ of sizes n − 1 and n, respectively.
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.