4,248 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
Composition of Permutation Representations of Triangle Groups
A triangle group is denoted by and has finite presentation We examine a method
for composition of permutation representations of a triangle group
that was used in Everitt's proof of Higman's 1968 conjecture
that every Fuchsian group has amongst its homomorphic images all but finitely
many alternating groups. We see that some of these compositions must give
imprimitive representations, and in particular situations, where the
representations being composed are all equivalent copies of an alternating
group in the same degree, we can give a complete description of the structure
of the composition of the representations. This article contains the main
results of the author's PhD thesis
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