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

Michael Larsen, Jay Taylor, Pham Huu Tiep

Research output: Contribution to journalArticlepeer-review

Abstract

For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over Fq, then every element x of 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 qp-1q-1, then every non-scalar element x∈ 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.

Original languageEnglish (US)
Article number47
JournalMathematische Zeitschrift
Volume303
Issue number2
DOIs
StatePublished - Feb 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

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