nLab
Plancherel measure

Contents

Context

Measure and probability theory

Representation theory

Contents

Idea

A probability distribution on Young diagrams.

Definition

From an answer by Vadim Alekseev?:

Suppose GG 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 λ:GU(L 2(G,Haar))\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)L^2(G) as a direct sum of irreducible reprenentations (this already fails for G=G=\mathbb{Z}), it is always possible to decompose it as a direct integral? of irreducible representations (which are parametrised by the unitary dual? G^\widehat G of GG). Now, if GG is unimodular and type I, the direct integral decomposition (with respect to both left and right actions of GG) is as follows:

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

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

f 2 2= G^π(f) HS 2dμ(π),fL 1(G,Haar)L 2(G,Haar). \|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 GG is a unimodular second countable Type I group, then the Plancherel measure is the unique measure μ\mu such that

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

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

When GG 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 GG 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 GG 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 GG. Suppose GG satisfies the same hypotheses as above and Δ:¯\Delta : \mathcal{M} \to \mathcal{M} \overline{\otimes} \mathcal{M} is the comultiplication on \mathcal{M} given by λ(s)λ(s)λ(s)\lambda(s) \mapsto \lambda(s) \otimes \lambda(s). Then the Plancherel trace is the unique normal semifinite trace τ\tau on \mathcal{M} such that

τ((φid)(Δ(a)))=τ(a)\tau((\varphi \otimes \mathrm{id}) (\Delta(a))) = \tau(a)

for all a τ +a \in \mathcal{M}_\tau^+ and φ *\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 λ\lambda a partition/Young diagram, its Plancherel probability is (see at hook length formula):

p λ Pl(dim(S (λ))) 2n!=n!1irows(λ)1jλ i1(hook λ(i,j)) 2, 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

Properties

With respect to the Plancherel measure on Part(n)Part(n) and in the limit of large nn \to \infty, the logarithm of dim(S (λ))=|sYTableaux λ|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 λ\lambda) on the value

ln(|sYTableaux λ|)c2n12ln(n!) ln \big( \left\vert sYTableaux_\lambda \right\vert \big) \;\sim\; \tfrac{c}{2} \sqrt{n} - \tfrac{1}{2}\ln(n!)

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

limnp Pl({λPart(n)|2nln|sYTableaux λ|n!c<ϵ})=1. \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))

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:

In relation to matrix models:

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

Last revised on July 6, 2021 at 10:51:48. See the history of this page for a list of all contributions to it.