Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture
| Autor: | Larsen, Michael, Taylor, Jay, Tiep, Pham |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Math. Z. 303 (2023), no.2, Paper No. 47, 45 pp |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00209-022-03193-3 |
| Popis: | For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of $\le k$ 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 $\frac{q^p-1}{q-1}$, then every non-scalar element $x \in \operatorname{SL}_p(q)$ 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. Comment: 40 pages, additional details for Theorem 3.1 available in the source |
| Databáze: | arXiv |
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