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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

   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 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 . Now, general theory tells us that while it’s not always possible to decompose 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 of ). Now, if is unimodular and type I, the direct integral decomposition (with respect to both left and right actions of ) is as follows:

   where , and its understanding requires, in particular, to determine the measure on such that the above becomes an isometric isomorphism. The unique measure with this property is called the Plancherel measure of (associated to a given Haar measure). Equivalently, it’s the unique measure such that

   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 is a unimodular second countable Type I group, then the Plancherel measure is the unique measure such that

   for every . This appears as Theorem 18.8.2 in Dixmier’s book on -algebras.

   When 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 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 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 generated by the left-regular representation of . Suppose satisfies the same hypotheses as above and is the comultiplication on given by . Then the Plancherel trace is the unique normal semifinite trace on such that

   for all and . 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