# 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 ${\displaystyle G}$, 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.

## Definition for finite groups

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

${\displaystyle \mu (\pi )={\frac {({\mathrm {dim} }\,\pi )^{2}}{|G|}},}$

where ${\displaystyle \pi \in G^{\wedge }}$, and ${\displaystyle {\mathrm {dim} }\pi }$ denotes the dimension of the irreducible representation ${\displaystyle \pi }$. [1]

## Definition on the symmetric group ${\displaystyle S_{n}}$

An important special case is the case of the finite symmetric group ${\displaystyle S_{n}}$, where ${\displaystyle n}$ is a positive integer. For this group, the set ${\displaystyle S_{n}^{\wedge }}$ of irreducible representations is in natural bijection with the set of integer partitions of ${\displaystyle n}$. For an irreducible representation associated with an integer partition ${\displaystyle \lambda }$, its dimension is known to be equal to ${\displaystyle f^{\lambda }}$, the number of standard Young tableaux of shape ${\displaystyle \lambda }$, 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

${\displaystyle \mu (\lambda )={\frac {(f^{\lambda })^{2}}{n!}}.}$ [2]

The fact that those probabilities sum up to 1 follows from the combinatorial identity

${\displaystyle \sum _{\lambda \vdash n}(f^{\lambda })^{2}=n!,}$

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 ${\displaystyle \sigma }$. 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 ${\displaystyle S_{n}}$.

### Connection to longest increasing subsequence

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

${\displaystyle L(\sigma )=\lambda _{1},\,}$

where ${\displaystyle \lambda _{1}}$ denotes the length of the first row of ${\displaystyle \lambda }$. Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of ${\displaystyle \lambda }$ is exactly the Plancherel measure on ${\displaystyle S_{n}}$. So, to understand the behavior of ${\displaystyle L(\sigma )}$, it is natural to look at ${\displaystyle \lambda _{1}}$ with ${\displaystyle \lambda }$ chosen according to the Plancherel measure in ${\displaystyle S_{n}}$, since these two random variables have the same probability distribution. [3]

### Poissonized Plancherel measure

Plancherel measure is defined on ${\displaystyle S_{n}}$ for each integer ${\displaystyle n}$. In various studies of the asymptotic behavior of ${\displaystyle L(\sigma )}$ as ${\displaystyle n\rightarrow \infty }$, it has proved useful [4] to extend the measure to a measure, called the Poissonized Plancherel measure, on the set ${\displaystyle {\mathcal {P}}^{*}}$ of all integer partitions. For any ${\displaystyle \theta >0}$, the Poissonized Plancherel measure with parameter ${\displaystyle \theta }$ on the set ${\displaystyle {\mathcal {P}}^{*}}$ is defined by

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

### Plancherel growth process

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

${\displaystyle p(\nu ,\lambda )={\mathbb {P} }(\lambda ^{(n)}=\lambda ~|~\lambda ^{(n-1)}=\nu )={\frac {f^{\lambda }}{nf^{\nu }}},}$

for any given Young diagrams ${\displaystyle \nu }$ and ${\displaystyle \lambda }$ of sizes n − 1 and n, respectively. [5]

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 ${\displaystyle \lambda ^{(n)}}$ in this walk coincides with the Plancherel measure on ${\displaystyle S_{n}}$. [6]

## 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.

## References

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