Sharp character bounds for symmetric groups in terms of partition length

TL;DR

This paper establishes a precise upper bound for the absolute value of irreducible characters of symmetric groups based on the number of disjoint cycles in an element, with implications for character bounds in related algebraic groups.

Contribution

It provides a sharp, explicit upper bound for character values in symmetric groups in terms of the cycle structure, extending to unipotent characters of $SL_n(q)$.

Findings

The bound |hi(g)| q k! for elements with k cycles.

The bound is sharp and attained for fixed k.

Implications for character bounds in algebraic groups like SL_n(q).

Abstract

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for fixed $k$ and varying $n$, $\chi$, and $g$. This implies a sharp upper bound of $k!$ for unipotent character values of $SL_n(q)$ at regular semisimple elements with characteristic polynomial $P(t)=P_1(t)\cdots P_k(t)$, where the $P_i$ are irreducible over $F_q[t]$.