Limit Values of Character Sums in Frobenius Formula of Three Permutations

TL;DR

This paper investigates the asymptotic limits of character sums in the Frobenius formula for three conjugacy classes of symmetric groups, revealing specific convergence behaviors based on cycle structures.

Contribution

It establishes new asymptotic results for character sums in symmetric groups with particular cycle configurations, advancing understanding of the Frobenius formula in these cases.

Findings

Character sum converges to 2 when classes have no cycles of lengths 1, 2, or 3.

Character sum converges to 2e^{-H^2} when two classes have H√n fixed points.

Results lead to corollaries and conjectures related to the Hurwitz problem.

Abstract

We study the asymptotic behavior of the character sums appearing in the Frobenius formula for three conjugacy classes of symmetric groups. We show that if all three conjugacy classes contain no cycles of lengths $1$, $2$, or $3$, then the character sum converges to $2$. On the other hand, if two of the conjugacy classes contain $H\sqrt{n}$ fixed points while all other cycle lengths in all three conjugacy classes are large, then the character sum converges to $2e^{-H^2}$. As consequences of these results, we obtain several corollaries and propose conjectures related to the Hurwitz problem with three branching points on the sphere.