Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913504770916352 |
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| author | Larsen, Michael Taylor, Jay Tiep, Pham |
| author_facet | Larsen, Michael Taylor, Jay Tiep, Pham |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_09262 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture Larsen, Michael Taylor, Jay Tiep, Pham Representation Theory 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. |
| title | Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2204.09262 |