Asymptotics of characters of symmetric groups, genus expansion and free probability

17Citations
Citations of this article
14Readers
Mendeley users who have this article in their library.

Abstract

The convolution of indicators of two conjugacy classes on the symmetric group Sq 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 Sq 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. © 2006 Elsevier B.V. All rights reserved.

Cite

CITATION STYLE

APA

Śniady, P. (2006). Asymptotics of characters of symmetric groups, genus expansion and free probability. Discrete Mathematics, 306(7), 624–665. https://doi.org/10.1016/j.disc.2006.02.004

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free