Symmetric groups and random matrices

Details Symmetric groups and random matrices Symmetric groups and random matrices

2003-01-26

arxiv.org/abs/math/0301299v3

Mathematics

Combinatorics

This paper has been withdrawn by the author because preprints math.CO/0301299 and math.CO/0304275 were superceded by the paper

Scientific paper

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 and we show that it very closely related to the combinatorial approach to random matrices. Our formulas are exact (in a sense that they hold not only asymptotically for large q). This result has many interesting applications, for example it allows to find precise asymptotics of characters of large symmetric groups and asymptotics of the Plancherel measure on Young diagrams.

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