Difference between revisions of "Plancherel measure"

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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.|coauthors=Okounkov, A.|title=Asymptotics of Plancherel measures for symmetric groups|journal=J. Amer. Math. Soc.|year=2000|series=13:491–515}}</ref>
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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.|coauthors=Shepp, L. A.|title=A variational problem for random Young tableaux|journal=Adv. Math.|year=1977|series=26:206–222}}</ref>
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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.|coauthors=Deift, P.; 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
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'''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]]
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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''&nbsp;&minus;&nbsp;1 and&nbsp;''n'', respectively. <ref name=Vershik>{{cite journal|last=Vershik|first=A. M.|coauthors=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>
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for any given Young diagrams <math>\nu</math> and <math>\lambda</math> of sizes ''n''&nbsp;&minus;&nbsp;1 and&nbsp;''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>  
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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.
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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

[2]

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

for all . [2]

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 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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