# Difference between revisions of "Plancherel measure"

en>Anrnusna m (→Definition on the symmetric group <math>S_n</math>: journal name, replaced: Ann. Math. → Annals of Mathematics using AWB) |
(→Plancherel growth process: added link to Young tableaux) |
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:<math>\mu(\pi) = \frac{(\mathrm{dim}\,\pi)^2}{|G|},</math> | :<math>\mu(\pi) = \frac{(\mathrm{dim}\,\pi)^2}{|G|},</math> | ||

− | where <math>\pi\in G^\wedge</math>, and <math>\mathrm{dim}\pi</math> denotes the dimension of the irreducible representation <math>\pi</math>. <ref name=Borodin>{{cite journal|last=Borodin|first=A.| | + | where <math>\pi\in G^\wedge</math>, and <math>\mathrm{dim}\pi</math> denotes the dimension of the irreducible representation <math>\pi</math>. <ref name=Borodin>{{cite journal|last=Borodin|first=A.|author2=Okounkov, A. |title=Asymptotics of Plancherel measures for symmetric groups|journal=J. Amer. Math. Soc.|year=2000|series=13:491–515}}</ref> |

==Definition on the symmetric group <math>S_n</math>== | ==Definition on the symmetric group <math>S_n</math>== | ||

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:<math>L(\sigma) = \lambda_1, \,</math> | :<math>L(\sigma) = \lambda_1, \,</math> | ||

− | where <math>\lambda_1</math> denotes the length of the first row of <math>\lambda</math>. Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of <math>\lambda</math> is exactly the Plancherel measure on <math>S_n</math>. So, to understand the behavior of <math>L(\sigma)</math>, it is natural to look at <math>\lambda_1</math> with <math>\lambda</math> chosen according to the Plancherel measure in <math>S_n</math>, since these two random variables have the same probability distribution. <ref name=Logan>{{cite journal|last=Logan|first=B. F.| | + | where <math>\lambda_1</math> denotes the length of the first row of <math>\lambda</math>. Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of <math>\lambda</math> is exactly the Plancherel measure on <math>S_n</math>. So, to understand the behavior of <math>L(\sigma)</math>, it is natural to look at <math>\lambda_1</math> with <math>\lambda</math> chosen according to the Plancherel measure in <math>S_n</math>, since these two random variables have the same probability distribution. <ref name=Logan>{{cite journal|last=Logan|first=B. F.|author2=Shepp, L. A. |title=A variational problem for random Young tableaux|journal=Adv. Math.|year=1977|series=26:206–222}}</ref> |

=== Poissonized Plancherel measure === | === Poissonized Plancherel measure === | ||

− | '''Plancherel measure''' is defined on <math>S_n</math> for each integer <math>n</math>. In various studies of the asymptotic behavior of <math>L(\sigma)</math> as <math>n \rightarrow \infty</math>, it has proved useful <ref name=BDJ>{{cite journal|last=Baik|first=J.| | + | '''Plancherel measure''' is defined on <math>S_n</math> for each integer <math>n</math>. In various studies of the asymptotic behavior of <math>L(\sigma)</math> as <math>n \rightarrow \infty</math>, it has proved useful <ref name=BDJ>{{cite journal|last=Baik|first=J.|author2=Deift, P. |author3=Johansson, K. |title=On the distribution of the length of the longest increasing subsequence of random permutations|journal=J. Amer. Math. Soc.|year=1999|series=12:1119–1178}}</ref> to extend the measure to a measure, called the '''Poissonized Plancherel measure''', on the set <math>\mathcal{P}^*</math> of all integer partitions. For any <math>\theta > 0</math>, the '''Poissonized Plancherel measure with parameter <math>\theta</math>''' on the set <math>\mathcal{P}^*</math> is defined by |

:<math>\mu_\theta(\lambda) = e^{-\theta}\frac{\theta^{|\lambda|}(f^\lambda)^2}{(|\lambda|!)^2},</math> | :<math>\mu_\theta(\lambda) = e^{-\theta}\frac{\theta^{|\lambda|}(f^\lambda)^2}{(|\lambda|!)^2},</math> | ||

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=== Plancherel growth process === | === Plancherel growth process === | ||

− | The '''Plancherel growth process''' is a random sequence of Young diagrams <math>\lambda^{(1)} = (1),~\lambda^{(2)},~\lambda^{(3)},~\ldots,</math> such that each <math>\lambda^{(n)}</math> is a random Young diagram of order <math>n</math> whose probability distribution is the ''n''th Plancherel measure, and each successive <math>\lambda^{(n)}</math> is obtained from its predecessor <math>\lambda^{(n-1)}</math> by the addition of a single box, according to the [[transition probability]] | + | The '''Plancherel growth process''' is a random sequence of [[Young tableaux|Young diagrams]] <math>\lambda^{(1)} = (1),~\lambda^{(2)},~\lambda^{(3)},~\ldots,</math> such that each <math>\lambda^{(n)}</math> is a random Young diagram of order <math>n</math> whose probability distribution is the ''n''th Plancherel measure, and each successive <math>\lambda^{(n)}</math> is obtained from its predecessor <math>\lambda^{(n-1)}</math> by the addition of a single box, according to the [[transition probability]] |

:<math>p(\nu, \lambda) = \mathbb{P}(\lambda^{(n)}=\lambda~|~\lambda^{(n-1)}=\nu) = \frac{f^{\lambda}}{nf^{\nu}},</math> | :<math>p(\nu, \lambda) = \mathbb{P}(\lambda^{(n)}=\lambda~|~\lambda^{(n-1)}=\nu) = \frac{f^{\lambda}}{nf^{\nu}},</math> | ||

− | for any given Young diagrams <math>\nu</math> and <math>\lambda</math> of sizes ''n'' − 1 and ''n'', respectively. <ref name=Vershik>{{cite journal|last=Vershik|first=A. M.| | + | for any given Young diagrams <math>\nu</math> and <math>\lambda</math> of sizes ''n'' − 1 and ''n'', respectively. <ref name=Vershik>{{cite journal|last=Vershik|first=A. M.|author2=Kerov, S. V. |title=The asymptotics of maximal and typical dimensions irreducible representations of the symmetric group|journal=Funct. Anal. Appl.|year=1985|series=19:21–31}}</ref> |

− | 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 <math>\lambda^{(n)}</math> in this walk coincides with the '''Plancherel measure''' on <math>S_n</math>. <ref name=Kerov>{{cite journal|last=Kerov|first=S.|title=A differential model of growth of Young diagrams|journal=Proceedings of St.Petersburg Mathematical Society|year=1996}}</ref> | + | 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 <math>\lambda^{(n)}</math> in this walk coincides with the '''Plancherel measure''' on <math>S_n</math>. <ref name=Kerov>{{cite journal|last=Kerov|first=S.|title=A differential model of growth of Young diagrams|journal=Proceedings of St.Petersburg Mathematical Society|year=1996}}</ref> |

==Compact groups== | ==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. | + | 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== | ==Abelian groups== |

## Latest revision as of 14:52, 3 October 2014

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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