In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group
In some cases the term Plancherel measure is applied specifically in the context of the group
It is named after the Swiss mathematician Michel Plancherel for his work in representation theory.
be a finite group, we denote the set of its irreducible representations by
The corresponding Plancherel measure over the set
denotes the dimension of the irreducible representation
of irreducible representations is in natural bijection with the set of integer partitions of
For an irreducible representation associated with an integer partition
, 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.
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
denote the length of a longest increasing subsequence of a random permutation
denote the shape of the corresponding Young tableaux related to
Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of
, since these two random variables have the same probability distribution.
[3] Plancherel measure is defined on
, the Poissonized Plancherel measure with parameter
[2] The Plancherel growth process is a random sequence of Young diagrams
is a random Young diagram of order
whose probability distribution is the nth Plancherel measure, and each successive
by the addition of a single box, according to the transition probability for any given Young diagrams
[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] 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.
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.
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.