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

DOI10.1007/S00209-022-03193-3arXiv2204.09262WikidataQ123346324 ScholiaQ123346324MaRDI QIDQ6396946

Jay Taylor, Pham Huu Tiep, Michael Larsen

Publication date: 20 April 2022

Abstract: For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $ginoperatorname{SL}_n(q)$ is the product of $lek$ pairwise distinct monic irreducible polynomials over $mathbb{F}_q$, then every element $x$ of $operatorname{SL}_n(q)$ 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 $frac{q}^p-1q-1$, then every non-scalar element $xinoperatorname{SL}_p(q)$ 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.

Mathematics Subject Classification ID

Conjugacy classes for groups (20E45) Linear algebraic groups over finite fields (20G40) Representation theory for linear algebraic groups (20G05) Finite simple groups and their classification (20D05) Representations of finite groups of Lie type (20C33)

This page was built for publication: Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6396946)