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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something <a href="https://ncatlab.org/nlab/revision/Plancherel+measure/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Plancherel+measure">current</a>

   starting something

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's: * Suvankar Dutta, Debangshu Mukherjee, Neetu, Sanhita Parihar, *A Unitary Matrix Model for q-deformed Plancherel Growth* ([arXiv:2105.09342](https://arxiv.org/abs/2105.09342)) > In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure. <a href="https://ncatlab.org/nlab/revision/diff/Plancherel+measure/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Plancherel+measure/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Plancherel+measure">current</a>

   added pointer to today’s:

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

   In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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 * [[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](https://doi.org/10.1007/BF01086021)) <a href="https://ncatlab.org/nlab/revision/diff/Plancherel+measure/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Plancherel+measure/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Plancherel+measure">current</a>

   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

   • 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)

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded an actual definition: ## Definition From a [[MathOverflow]] [answer](https://mathoverflow.net/questions/284756/characterizing-the-plancherel-measure/284762#284762) 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 $$ \|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi),\quad f\in L^1(G,\mathrm{Haar})\cap L^2(G,\mathrm{Haar}). $$ <a href="https://ncatlab.org/nlab/revision/diff/Plancherel+measure/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Plancherel+measure/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Plancherel+measure">current</a>

   Added an actual definition:

   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

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

   diff, v7, current

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAnd another one: From a [[MathOverflow]] [answer](https://mathoverflow.net/questions/284756/characterizing-the-plancherel-measure/284762#284762) 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. <a href="https://ncatlab.org/nlab/revision/diff/Plancherel+measure/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Plancherel+measure/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Plancherel+measure">current</a>

   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.

   diff, v7, current