264 research outputs found
Asymptotics of characters of symmetric groups, genus expansion and free probability
The convolution of indicators of two conjugacy classes on the symmetric group
S_q is usually a complicated linear combination of indicators of many conjugacy
classes. Similarly, a product of the moments of the Jucys--Murphy element
involves many conjugacy classes with complicated coefficients. In this article
we consider a combinatorial setup which allows us to manipulate such products
easily: to each conjugacy class we associate a two-dimensional surface and the
asymptotic properties of the conjugacy class depend only on the genus of the
resulting surface. This construction closely resembles the genus expansion from
the random matrix theory. As the main application we study irreducible
representations of symmetric groups S_q for large q. We find the asymptotic
behavior of characters when the corresponding Young diagram rescaled by a
factor q^{-1/2} converge to a prescribed shape. The character formula (known as
the Kerov polynomial) can be viewed as a power series, the terms of which
correspond to two-dimensional surfaces with prescribed genus and we compute
explicitly the first two terms, thus we prove a conjecture of Biane.Comment: version 2: change of title; the section on Gaussian fluctuations was
moved to a subsequent paper [Piotr Sniady: "Gaussian fluctuations of
characters of symmetric groups and of Young diagrams" math.CO/0501112
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
Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
We find an explicit combinatorial interpretation of the coefficients of Kerov
character polynomials which express the value of normalized irreducible
characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,...
of the corresponding Young diagram. Our interpretation is based on counting
certain factorizations of a given permutation
Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
We study asymptotics of an irreducible representation of the symmetric group Sn corresponding to a balanced Young diagram λ (a Young diagram with at most View the MathML source rows and columns for some fixed constant C) in the limit as n tends to infinity
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks
AbstractFuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman's conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Î has (n!)ÎŒ+o(1) index n subgroups, where ÎŒ is the measure of Î, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of âalmost homogeneousâ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups
Gaussian fluctuations of Young diagrams and structure constants of Jack characters
In this paper, we consider a deformation of Plancherel measure linked to Jack
polynomials. Our main result is the description of the first and second-order
asymptotics of the bulk of a random Young diagram under this distribution,
which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the
first order asymptotics) and Kerov (for the second order asymptotics). This
gives more evidence of the connection with Gaussian -ensemble, already
suggested by some work of Matsumoto.
Our main tool is a polynomiality result for the structure constant of some
quantities that we call Jack characters, recently introduced by Lassalle. We
believe that this result is also interested in itself and we give several other
applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of
this work, with significantly fewer results and a different title, is
available as arXiv:1201.180
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