Some Identities and Asymptotics for Characters of the Symmetric Group

Abstract

Ž w This paper is motivated by the problem arising from quantum gravity 9, x. 2 of counting combinatorial types of triangulations S of a Riemann surface X with given degrees of vertices. Let us color 2-simplexes of the barycentric subdivision S9 in black and white, so that adjacent simplexes are of different color, and consider a mapping p : X a P onto the Riemann sphere P which send white simplexes onto the northern hemisphere, black simplexes onto the southern hemisphere, and centers k-simplexes of S into an equatorial point y , k s 0, 1, 2. Then deg p s 3n, n is k the number of 2-simplexes in S, and p is unramified outside y , y , y . It 0 1 2 is easy to see that points of X over y and y have ramification index 2 1 2 Ž . respectively, 3 , while rmaification indices of points over y are just 0 degrees of vertices of S. So the problem of counting triangulations reduces to the problem of counting ramified coverings p : X a Y of Riemann surfaces with given ramification indices. In Section 2 we recall a connection between ramified coverings p : X a Y of a Riemann surface Y of genus g and irreducible characters x of the Y symmetric group S , n s deg p . The starting point is the following forn w x Ž w x. mula, essentially owing to Hurwitz 6 see also 10, 4, 2 :