Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture

Abstract

For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in {\mathrm{SL}}_n(q)$ is the product of $\le k$ pairwise distinct monic irreducible polynomials over ${\mathbb{F}}_q$, then every element $x$ of ${\mathrm{SL}}_n(q)$ of support at least $B$ is the product of two conjugates of $g$. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions $(p,q)$, in the special case that $n=p$ is prime, if $g$ has order $\frac{q^p-1}{q-1}$, then every non-scalar element $x\in {\mathrm{SL}}_p(q)$ is the product of two conjugates of $g$. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

40 pages, additional details for Theorem 3.1 available in the source

References in corpus (2)

• Character Levels and Character Bounds
• The combinatorics of ${\mathrm{GL}}_n$ generalized Gelfand--Graev characters