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| In [[mathematics]], '''Plancherel measure''' is a [[measure (mathematics)|measure]] defined on the set of [[irreducible representation|irreducible unitary representations]] of a [[locally compact group]] <math>G</math>, 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 <math>G</math> being the finite symmetric group <math>S_n</math> – see below. It is named after the Swiss mathematician [[Michel Plancherel]] for his work in [[representation theory]].
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| ==Definition for finite groups==
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| Let <math>G</math> be a [[finite group]], we denote the set of its [[irreducible representation]]s by <math>G^\wedge</math>. The corresponding '''Plancherel measure''' over the set <math>G^\wedge</math> is defined by
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| :<math>\mu(\pi) = \frac{(\mathrm{dim}\,\pi)^2}{|G|},</math>
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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.|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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| ==Definition on the symmetric group <math>S_n</math>==
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| An important special case is the case of the finite [[symmetric group]] <math>S_n</math>, where <math>n</math> is a positive integer. For this group, the set <math>S_n^\wedge</math> of irreducible representations is in natural bijection with the set of [[integer partitions]] of <math>n</math>. For an irreducible representation associated with an integer partition <math>\lambda</math>, its dimension is known to be equal to <math>f^\lambda</math>, the number of [[Young tableaux|standard Young tableaux]] of shape <math>\lambda</math>, 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
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| :<math>\mu(\lambda) = \frac{(f^\lambda)^2}{n!}.</math> <ref name=Johansson>{{cite journal|last=Johansson|first=K.|title=Discrete orthogonal polynomial ensembles and the Plancherel measure|journal=Annals of Mathematics|year=2001|series=153:259–296}}</ref>
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| The fact that those probabilities sum up to 1 follows from the combinatorial identity
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| :<math>\sum_{\lambda \vdash n}(f^\lambda)^2 = n!,</math>
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| which corresponds to the bijective nature of the [[Robinson–Schensted correspondence]].
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| ==Application==
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| '''Plancherel measure''' appears naturally in combinatorial and probabilistic problems, especially in the study of [[longest increasing subsequence]] of a random [[permutation]] <math>\sigma</math>. 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 <math>S_n</math>.
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| === Connection to longest increasing subsequence ===
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| Let <math>L(\sigma)</math> denote the length of a longest increasing subsequence of a random [[permutation]] <math>\sigma</math> in <math>S_n</math> chosen according to the uniform distribution. Let <math>\lambda</math> denote the shape of the corresponding [[Young tableau]]x related to <math>\sigma</math> by the [[Robinson–Schensted correspondence]]. Then the following identity holds:
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| :<math>L(\sigma) = \lambda_1, \,</math>
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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.|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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| === Poissonized Plancherel measure ===
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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.|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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| :<math>\mu_\theta(\lambda) = e^{-\theta}\frac{\theta^{|\lambda|}(f^\lambda)^2}{(|\lambda|!)^2},</math>
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| for all <math>\lambda \in \mathcal{P}^*</math>. <ref name=Johansson/> | |
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| === Plancherel growth process ===
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| 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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| :<math>p(\nu, \lambda) = \mathbb{P}(\lambda^{(n)}=\lambda~|~\lambda^{(n-1)}=\nu) = \frac{f^{\lambda}}{nf^{\nu}},</math>
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| 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.|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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| 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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| ==Compact groups==
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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.
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| ==Abelian groups==
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| 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.
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| ==Semisimple Lie groups==
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| The Plancherel measure for semisimple Lie groups was found by [[Harish-Chandra]]. The support is the set of [[tempered representation]]s, and in particular not all unitary representations need occur in the support.
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| ==References==
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| {{reflist}}
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| [[Category:Representation theory]]
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The adore for branded outfits and extras is not new, but men and females are just in appreciate with shopping branded objects for themselves.
This is the purpose, why the popularity and desire for designer outfits, footwear and purses have been escalating by just about every passing working day.�
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