330 research outputs found
Gaussian fluctuations of characters of symmetric groups and of Young diagrams
We study asymptotics of reducible representations of the symmetric groups S_q
for large q. We decompose such a representation as a sum of irreducible
components (or, alternatively, Young diagrams) and we ask what is the character
of a randomly chosen component (or, what is the shape of a randomly chosen
Young diagram). Our main result is that for a large class of representations
the fluctuations of characters (and fluctuations of the shape of the Young
diagrams) are asymptotically Gaussian; in this way we generalize Kerov's
central limit theorem. The considered class consists of representations for
which the characters almost factorize and this class includes, for example,
left-regular representation (Plancherel measure), tensor representations. This
class is also closed under induction, restriction, outer product and tensor
product of representations. Our main tool in the proof is the method of genus
expansion, well known from the random matrix theory.Comment: 37 pages; version 3: conceptual change in the proof
Harmonic analysis on the infinite symmetric group
Let S be the group of finite permutations of the naturals 1,2,... The subject
of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands
for the product of two copies of S while K is the diagonal subgroup in G. The
spherical dual to (G,K) (that is, the set of irreducible spherical unitary
representations) is an infinite-dimensional space. For such Gelfand pairs, the
conventional scheme of harmonic analysis is not applicable and it has to be
suitably modified.
We construct a compactification of S called the space of virtual
permutations. It is no longer a group but it is still a G-space. On this space,
there exists a unique G-invariant probability measure which should be viewed as
a true substitute of Haar measure. More generally, we define a 1-parameter
family of probability measures on virtual permutations, which are
quasi-invariant under the action of G.
Using these measures we construct a family {T_z} of unitary representations
of G depending on a complex parameter z. We prove that any T_z admits a unique
decomposition into a multiplicity free integral of irreducible spherical
representations of (G,K). Moreover, the spectral types of different
representations (which are defined by measures on the spherical dual) are
pairwise disjoint.
Our main result concerns the case of integral values of parameter z: then we
obtain an explicit decomposition of T_z into irreducibles. The case of
nonintegral z is quite different. It was studied by Borodin and Olshanski, see
e.g. the survey math.RT/0311369.Comment: AMS Tex, 80 pages, no figure
Asymptotics of characters of symmetric groups related to Stanley character formula
We prove an upper bound for characters of the symmetric groups. Namely, we
show that there exists a constant a>0 with a property that for every Young
diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr
\rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n,
c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors
needed to write \pi\in S_n as a product of transpositions. We also give uniform
estimates for the error term in the Vershik-Kerov's and Biane's character
formulas and give a new formula for free cumulants of the transition measure.Comment: Version 4: Change of title, shortened to 20 pages. Version 3: 24
pages, the title and the list of authors were changed. Version 2: 14 pages,
the title, abstract and the main result were changed. Version 1: 10 pages
(mistake in Lemma 7- which is false
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