Plancherel measure

In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group ${\displaystyle G}$, 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 ${\displaystyle G}$ being the finite symmetric group ${\displaystyle S_{n}}$ – see below. It is named after the Swiss mathematician Michel Plancherel for his work in representation theory.

Definition for finite groups

Let ${\displaystyle G}$ be a finite group, we denote the set of its irreducible representations by ${\displaystyle G^{\wedge }}$. The corresponding Plancherel measure over the set ${\displaystyle G^{\wedge }}$ is defined by

${\displaystyle \mu (\pi )={\frac {(\mathrm {dim} \,\pi )^{2}}{|G|}},}$

where ${\displaystyle \pi \in G^{\wedge }}$, and ${\displaystyle \mathrm {dim} \pi }$ denotes the dimension of the irreducible representation ${\displaystyle \pi }$. ^[1]

Definition on the symmetric group

An important special case is the case of the finite symmetric group ${\displaystyle S_{n}}$, where ${\displaystyle n}$ is a positive integer. For this group, the set ${\displaystyle S_{n}^{\wedge }}$ of irreducible representations is in natural bijection with the set of integer partitions of ${\displaystyle n}$. For an irreducible representation associated with an integer partition ${\displaystyle \lambda }$, its dimension is known to be equal to ${\displaystyle f^{\lambda }}$, the number of standard Young tableaux of shape ${\displaystyle \lambda }$, 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

${\displaystyle \mu (\lambda )={\frac {(f^{\lambda })^{2}}{n!}}.}$ ^[2]

The fact that those probabilities sum up to 1 follows from the combinatorial identity

${\displaystyle \sum _{\lambda \vdash n}(f^{\lambda })^{2}=n!,}$

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 ${\displaystyle \sigma }$. 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 ${\displaystyle S_{n}}$.

Connection to longest increasing subsequence

Let ${\displaystyle L(\sigma )}$ denote the length of a longest increasing subsequence of a random permutation ${\displaystyle \sigma }$ in ${\displaystyle S_{n}}$ chosen according to the uniform distribution. Let ${\displaystyle \lambda }$ denote the shape of the corresponding Young tableaux related to ${\displaystyle \sigma }$ by the Robinson–Schensted correspondence. Then the following identity holds:

${\displaystyle L(\sigma )=\lambda _{1},}$

where ${\displaystyle \lambda _{1}}$ denotes the length of the first row of ${\displaystyle \lambda }$. Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of ${\displaystyle \lambda }$ is exactly the Plancherel measure on ${\displaystyle S_{n}}$. So, to understand the behavior of ${\displaystyle L(\sigma )}$, it is natural to look at ${\displaystyle \lambda _{1}}$ with ${\displaystyle \lambda }$ chosen according to the Plancherel measure in ${\displaystyle S_{n}}$, since these two random variables have the same probability distribution. ^[3]

Poissonized Plancherel measure

Plancherel measure is defined on ${\displaystyle S_{n}}$ for each integer ${\displaystyle n}$. In various studies of the asymptotic behavior of ${\displaystyle L(\sigma )}$ as ${\displaystyle n\rightarrow \infty }$, it has proved useful ^[4] to extend the measure to a measure, called the Poissonized Plancherel measure, on the set ${\displaystyle {\mathcal {P}}^{*}}$ of all integer partitions. For any ${\displaystyle \theta >0}$, the Poissonized Plancherel measure with parameter ${\displaystyle \theta }$ on the set ${\displaystyle {\mathcal {P}}^{*}}$ is defined by

${\displaystyle \mu _{\theta }(\lambda )=e^{-\theta }{\frac {\theta ^{|\lambda |}(f^{\lambda })^{2}}{(|\lambda |!)^{2}}},}$

for all ${\displaystyle \lambda \in {\mathcal {P}}^{*}}$. ^[2]

Plancherel growth process

The Plancherel growth process is a random sequence of Young diagrams ${\displaystyle \lambda ^{(1)}=(1),~\lambda ^{(2)},~\lambda ^{(3)},~\ldots ,}$ such that each ${\displaystyle \lambda ^{(n)}}$ is a random Young diagram of order ${\displaystyle n}$ whose probability distribution is the nth Plancherel measure, and each successive ${\displaystyle \lambda ^{(n)}}$ is obtained from its predecessor ${\displaystyle \lambda ^{(n-1)}}$ by the addition of a single box, according to the transition probability

${\displaystyle p(\nu ,\lambda )=\mathbb {P} (\lambda ^{(n)}=\lambda ~|~\lambda ^{(n-1)}=\nu )={\frac {f^{\lambda }}{nf^{\nu }}},}$

for any given Young diagrams ${\displaystyle \nu }$ and ${\displaystyle \lambda }$ 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 ${\displaystyle \lambda ^{(n)}}$ in this walk coincides with the Plancherel measure on ${\displaystyle S_{n}}$. ^[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

1. Borodin, Alexei; Okounkov, Andrei; Olshanski, Grigori (2000). "Asymptotics of Plancherel measures for symmetric groups". Journal of the American Mathematical Society. 13:491–515. 13 (3): 481–515. doi: 10.1090/S0894-0347-00-00337-4 . S2CID   14183320.
2. Johansson, Kurt (2001). "Discrete orthogonal polynomial ensembles and the Plancherel measure". Annals of Mathematics. 153 (1): 259–296. arXiv: math/9906120 . doi:10.2307/2661375. JSTOR   2661375. S2CID   14120881.
3. Logan, B. F.; Shepp, L. A. (1977). "A variational problem for random Young tableaux". Advances in Mathematics . 26 (2): 206–222. doi: 10.1016/0001-8708(77)90030-5 .
4. Baik, Jinho; Deift, Percy; Johansson, Kurt (1999). "On the distribution of the length of the longest increasing subsequence of random permutations". Journal of the American Mathematical Society. 12:1119–1178. 12 (4): 1119–1178. doi: 10.1090/S0894-0347-99-00307-0 . S2CID   11355968.
5. Vershik, A. M.; Kerov, S. V. (1985). "The asymptotics of maximal and typical dimensions irreducible representations of the symmetric group". Funct. Anal. Appl. 19:21–31. doi: 10.1007/BF01086021 . S2CID   120927640.
6. Kerov, S. (1996). "A differential model of growth of Young diagrams". Proceedings of St.Petersburg Mathematical Society.