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 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
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.
For a partition/Young diagram, its Plancherel probability is (see at hook length formula):
where
denotes the complex irrep of the symmetric group that is labelled by via the representation theory of the symmetric group (the th Specht module)
denotes the hook length at the -box in the Young diagram .
With respect to the Plancherel measure on and in the limit of large , the logarithm of (see at hook length formula) is almost surely approximately constant (i.e. independent of ) on the value
for some (numerically: ), in that for all 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.