Plancherel measure
Encyclopedia
In mathematics
, Plancherel measure is a probability measure
defined on the set of irreducible representations of a finite group
. 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
.
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
, 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.
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
in 
chosen according to the uniform distribution. Let 
denote the shape of the corresponding Young tableau
x 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.
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
.
such that each 
is a random Young diagram of order 
whose probability distribution is the nth 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.
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 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Plancherel measure is a probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
defined on the set of irreducible representations of a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
. 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 PlancherelMichel Plancherel
Michel Plancherel was a Swiss mathematician. He was born in Bussy and obtained his diploma in mathematics from the University of Fribourg in 1907...
for his work in representation theory
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 studiesmodules over these abstract algebraic structures...
.
General definition
Let be a finite groupFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
, we denote the set of its irreducible representations by
. The corresponding Plancherel measure over the set is defined bywhere
, and denotes the dimension of the irreducible representation . Definition on the symmetric group 
An important special case is the case of the finite symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
, 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 subsequenceLongest increasing subsequence
The longest increasing subsequence problem is to find a subsequence of a given sequence in which the subsequence elements are in sorted order, lowest to highest, and in which the subsequence is as long as possible...
of a random permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
. 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 permutationPermutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
in chosen according to the uniform distribution. Let denote the shape of the corresponding Young tableauYoung tableau
In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
x 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. 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 byfor 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 nth Plancherel measure, and each successive is obtained from its predecessor by the addition of a single box, according to the transition probabilityfor 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
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
on Young's lattice
Young's lattice
In mathematics, Young's lattice is a partially ordered set and 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 representation theory of the symmetric group...
. It is not difficult to show that the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of
in this walk coincides with the Plancherel measure on .