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added pointer to today’s:
In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure.
added statement of
$\underset{n \to \infty}{\lim} p^{Pl} \left( \left\{ \lambda \in Part(n) \;\left\vert\; \tfrac{2}{\sqrt{n}} ln \frac {\left\vert sYTableaux_\lambda \right\vert} {\sqrt{n!}} - c \;\lt\; \epsilon \right. \right\} \right) \;=\; 1 \,.$from
Added an actual definition:
From a MathOverflow 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 ), it is always possible to decompose it as a direct integral of irreducible representations (which are parametrised by the unitary dual $G=\mathbb Z\widehat G$ 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
$\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi),\quad f\in L^1(G,\mathrm{Haar})\cap L^2(G,\mathrm{Haar}).$And another one:
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.
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