# Plancherel measure

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

## Definition for finite groups

Let be a finite group, we denote the set of its irreducible representations by . The corresponding **Plancherel measure** over the set is defined by

where , and denotes the dimension of the irreducible representation . ^{[1]}

## Definition on the symmetric group

An important special case is the case of the finite symmetric group , where is a positive integer. For this group, the set of irreducible representations is in natural bijection with the set of integer partitions of . For an irreducible representation associated with an integer partition , its dimension is known to be equal to , the number of standard Young tableaux of shape , 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

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

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 . 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 .

### Connection to longest increasing subsequence

Let denote the length of a longest increasing subsequence of a random permutation in chosen according to the uniform distribution. Let denote the shape of the corresponding Young tableaux related to by the Robinson–Schensted correspondence. Then the following identity holds:

where denotes the length of the first row of . Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of is exactly the Plancherel measure on . So, to understand the behavior of , it is natural to look at with chosen according to the Plancherel measure in , since these two random variables have the same probability distribution. ^{[3]}

### Poissonized Plancherel measure

**Plancherel measure** is defined on for each integer . In various studies of the asymptotic behavior of as , it has proved useful ^{[4]} to extend the measure to a measure, called the **Poissonized Plancherel measure**, on the set of all integer partitions. For any , the **Poissonized Plancherel measure with parameter ** on the set is defined by

### Plancherel growth process

The **Plancherel growth process** is a random sequence of Young diagrams such that each is a random Young diagram of order whose probability distribution is the *n*th Plancherel measure, and each successive is obtained from its predecessor by the addition of a single box, according to the transition probability

for any given Young diagrams and 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 in this walk coincides with the **Plancherel measure** on . ^{[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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