Plancherel measure

In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group formula_1, 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 formula_1 being the finite symmetric group formula_3 – see below. It is named after the Swiss mathematician Michel Plancherel for his work in representation theory.
Definition for finite groups.
Let formula_1 be a finite group, we denote the set of its irreducible representations by formula_5. The corresponding Plancherel measure over the set formula_5 is defined by
where formula_8, and formula_9 denotes the dimension of the irreducible representation formula_10. 
Definition on the symmetric group formula_3.
An important special case is the case of the finite symmetric group formula_3, where formula_13 is a positive integer. For this group, the set formula_14 of irreducible representations is in natural bijection with the set of integer partitions of formula_13. For an irreducible representation associated with an integer partition formula_16, its dimension is known to be equal to formula_17, the number of standard Young tableaux of shape formula_16, 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 formula_21. 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 formula_3. 
Connection to longest increasing subsequence.
Let formula_23 denote the length of a longest increasing subsequence of a random permutation formula_21 in formula_3 chosen according to the uniform distribution. Let formula_16 denote the shape of the corresponding Young tableaux related to formula_21 by the Robinson–Schensted correspondence. Then the following identity holds:
where formula_29 denotes the length of the first row of formula_16. Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of formula_16 is exactly the Plancherel measure on formula_3. So, to understand the behavior of formula_23, it is natural to look at formula_29 with formula_16 chosen according to the Plancherel measure in formula_3, since these two random variables have the same probability distribution. 
Poissonized Plancherel measure.
Plancherel measure is defined on formula_3 for each integer formula_13. In various studies of the asymptotic behavior of formula_23 as formula_40, it has proved useful to extend the measure to a measure, called the Poissonized Plancherel measure, on the set formula_41 of all integer partitions. For any formula_42, the Poissonized Plancherel measure with parameter formula_43 on the set formula_41 is defined by
for all formula_46. 
Plancherel growth process.
The Plancherel growth process is a random sequence of Young diagrams formula_47 such that each formula_48 is a random Young diagram of order formula_13 whose probability distribution is the "n"th Plancherel measure, and each successive formula_48 is obtained from its predecessor formula_51 by the addition of a single box, according to the transition probability
for any given Young diagrams formula_53 and formula_16 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 formula_48 in this walk coincides with the Plancherel measure on formula_3. 
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.