nLab Plancherel measure

Idea

A probability distribution on Young diagrams.

Definition

From an answer by Vadim Alekseev?:

Suppose $G$ is a locally compact group. What one ultimately wants to study is (upon fixing a Haar measure in the noncompact case) the left regular representation $\lambda\colon G\to U(L^2(G,\mathrm{Haar}))$. Now, general theory tells us that while it’s not always possible to decompose $L^2(G)$ as a direct sum of irreducible reprenentations (this already fails for $G=\mathbb{Z}$), it is always possible to decompose it as a direct integral of irreducible representations (which are parametrised by the unitary dual? $\widehat G$ of $G$). Now, if $G$ is unimodular and type I, the direct integral decomposition (with respect to both left and right actions of $G$) is as follows:

$$L^2(G) \cong \int_{\widehat G} H_\pi\,d\mu(\pi),$$

where $H_\pi = \pi\otimes \pi^*$, and its understanding requires, in particular, to determine the measure $\mu$ on $\widehat G$ such that the above becomes an isometric isomorphism. The unique measure with this property is called the Plancherel measure of $G$ (associated to a given Haar measure). Equivalently, it’s the unique measure such that

$$\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi),\quad f\in L^1(G,\mathrm{Haar})\cap L^2(G,\mathrm{Haar}).$$

From a MathOverflow answer by Cameron Zwarich?:

If $G$ is a unimodular second countable Type I group, then the Plancherel measure is the unique measure $\mu$ such that

$$\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi).$$

for every $f \in \mathrm{L}^1(G) \cap \mathrm{L}^2(G)$. This appears as Theorem 18.8.2 in Dixmier’s book on $C^*$-algebras.

When $G$ is not unimodular, the question becomes more complicated, because the Plancherel measure needs to be twisted by a section of a line bundle; see the paper of Duflo-Moore on the subject for the gory details. When $G$ is not second countable, I do not know of a published result; the technical details of direct integral theory are more difficult in this case and not standard. When $G$ is not Type I, the decomposition of the left regular representation into irreducibles is no longer unique, and some of the operators on the right-hand side of the formula will fail to have finite Hilbert-Schmidt norm.

The closest analogue to the definition of a Haar measure on abelian locally compact groups as a left-invariant Radon measure is the characterization of the Plancherel measure as a unique co-invariant trace (or weight) on the von Neumann algebra $\mathcal{M}$ generated by the left-regular representation of $G$. Suppose $G$ satisfies the same hypotheses as above and $\Delta : \mathcal{M} \to \mathcal{M} \overline{\otimes} \mathcal{M}$ is the comultiplication on $\mathcal{M}$ given by $\lambda(s) \mapsto \lambda(s) \otimes \lambda(s)$. Then the Plancherel trace is the unique normal semifinite trace $\tau$ on $\mathcal{M}$ such that

$$\tau((\varphi \otimes \mathrm{id}) (\Delta(a))) = \tau(a)$$

for all $a \in \mathcal{M}_\tau^+$ and $\varphi \in \mathcal{M}_*$. A similar characterization holds for the Plancherel weight of an arbitrary locally compact group, or for the Haar weight of a locally compact quantum group. For proofs, see volume 2 of Takesaki or any of the literature on von Neumann algebraic quantum groups.

Definition for symmetric groups

For $\lambda$ a partition/Young diagram, its Plancherel probability is (see at hook length formula):

$$p^{Pl}_\lambda \;\coloneqq\; \frac { \big( dim(S^{(\lambda)}) \big)^2 } {n!} \;=\; n! \underset{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } }{\prod} \frac{1}{ \big(\ell hook_\lambda(i,j)\big)^2 } \,,$$

where

• $S^{(\lambda)}$ denotes the complex irrep of the symmetric group that is labelled by $\lambda$ via the representation theory of the symmetric group (the $\lambda$th Specht module)

• $\ell hook_\lambda(i,j)$ denotes the hook length at the $(i,j)$-box in the Young diagram $\lambda$.

Properties

With respect to the Plancherel measure on $Part(n)$ and in the limit of large $n \to \infty$, the logarithm of $dim\big( S^{(\lambda)}\big) = \left\vert sYTableaux_\lambda \right\vert$ (see at hook length formula) is almost surely approximately constant (i.e. independent of $\lambda$) on the value

$$ln \big( \left\vert sYTableaux_\lambda \right\vert \big) \;\sim\; \tfrac{c}{2} \sqrt{n} - \tfrac{1}{2}\ln(n!)$$

for some $c \in \mathbb{R}$ (numerically: $c \gt 1.8$), in that for all $\epsilon \in \mathbb{R}_+$ we have

$$\underset{n \to \infty}{\lim} p^{Pl} \left( \left\{ \lambda \in Part(n) \;\left\vert\; \tfrac{2}{\sqrt{n}} ln \frac {\left\vert sYTableaux_\lambda \right\vert} {\sqrt{n!}} - c \;\lt\; \epsilon \right. \right\} \right) \;=\; 1 \,.$$

(Vershik & Kerov 85, p. 22 (2 of 11))

Related concepts

• Schur-Weyl measure

• hook length formula

• asymptotic representation theory

References

• Anatoly Vershik, Sergei Kerov, Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group, Functional Analysis and Its Applications volume 19, pages 21–31 (1985) (doi:10.1007/BF01086021)

• Maciej Dołęga, Central limit theorem for random Young diagrams with respect to Jack measure 2014 (pdf)

See also:

• Wikipedia, Plancherel measure

• Anatoly Vershik, Two lectures on the asymptotic representation theory and statistics of Young diagrams, In: Vershik A.M., Yakubovich Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics Lecture Notes in Mathematics, vol 1815. Springer 2003 (doi:10.1007/3-540-44890-X_7)

• G. Olshanski, Asymptotic representation theory, Lecture notes 2009-2010 (webpage, pdf 1, pdf 2)

In relation to matrix models:

• Suvankar Dutta, Debangshu Mukherjee, Neetu, Sanhita Parihar, A Unitary Matrix Model for q-deformed Plancherel Growth (arXiv:2105.09342)