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Uniform Character Bounds for Finite Classical Groups

Michael Larsen, Pham Huu Tiep

Abstract

For every finite quasisimple group of Lie type $G$, every irreducible character $χ$ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|χ(g)|/χ(1)$ with exponent linear in $\log_{|G|} |g^G|$, or, equivalently, in the ratio of the support of $g$ to the rank of $G$. We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic $2$, and some other infinite families of orthogonal and unitary groups

Uniform Character Bounds for Finite Classical Groups

Abstract

For every finite quasisimple group of Lie type , every irreducible character of , and every element of , we give an exponential upper bound for the character ratio with exponent linear in , or, equivalently, in the ratio of the support of to the rank of . We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic , and some other infinite families of orthogonal and unitary groups
Paper Structure (12 sections, 47 theorems, 326 equations)

This paper contains 12 sections, 47 theorems, 326 equations.

Table of Contents

  1. Introduction
  2. Counting lemmas
  3. Probabilistic lemmas
  4. Character bounds for elements with large support
  5. Further bootstrapping and uniform character bounds
  6. Support vs. class size, and proof of Theorem \ref{['main']}
  7. Squares of conjugacy classes and Thompson's conjecture
  8. Further applications
  9. Mixing time of random walks on Cayley graphs
  10. McKay graphs and products of irreducible characters
  11. Power word maps on simple groups
  12. Fibers of product morphisms on semisimple algebraic groups

Key Result

Theorem A

There exists an absolute constant $c>0$ such that for all finite quasisimple groups $G$ of Lie type, irreducible characters $\chi$ of $G$, and elements $g\in G$, we have

Theorems & Definitions (94)

  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 84 more