geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
A probability distribution on Young diagrams.
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:
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
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
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
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.
For $\lambda$ a partition/Young diagram, its Plancherel probability is (see at hook length formula):
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 $\lambda$.
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
for some $c \in \mathbb{R}$ (numerically: $c \gt 1.8$), in that for all $\epsilon \in \mathbb{R}_+$ we have
(Vershik & Kerov 85, p. 22 (2 of 11))
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:
Last revised on July 6, 2021 at 14:51:48. See the history of this page for a list of all contributions to it.