In
mathematics, Plancherel measure is a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
defined on the set of
irreducible unitary representations of a
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
, 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
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
.
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
.
Definition on the symmetric group
An important special case is the case of the finite
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, 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
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same parti ...
of
. For an irreducible representation associated with an integer partition
, its dimension is known to be equal to
, the number of
standard Young tableaux In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and ...
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 In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has various descriptions, all of which are of algorithmic nature, it has many rem ...
.
Application
Plancherel measure appears naturally in combinatorial and probabilistic problems, especially in the study of
longest increasing subsequence In computer science, the longest increasing subsequence problem is to find a subsequence of a given sequence in which the subsequence's elements are in sorted order, lowest to highest, and in which the subsequence is as long as possible. This subseq ...
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 tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...
x related to
by the
Robinson–Schensted correspondence In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has various descriptions, all of which are of algorithmic nature, it has many rem ...
. 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.
Poissonized Plancherel measure
Plancherel measure is defined on
for each integer
. In various studies of the asymptotic behavior of
as
, it has proved useful
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
:
for all
.
[
]
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
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
:
for any given Young diagrams and 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
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
on Young's lattice
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the representation theory of the symmetric ...
. It is not difficult to show that the probability distribution of in this walk coincides with the Plancherel measure on .
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 In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
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
Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early life
Harish-Chandra ...
. The support is the set of tempered representation In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space
:''L''2+ε(''G'')
for any ε > 0.
Formulation
This condition, as just g ...
s, and in particular not all unitary representations need occur in the support.
References
{{reflist
Representation theory