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Asymptotics of the powers in finite reductive groups

Amit Kulshrestha, Rijubrata Kundu, Anupam Singh

Abstract

Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$.

Asymptotics of the powers in finite reductive groups

Abstract

Let be a connected reductive group defined over . Fix an integer , and consider the power map on . We denote the image of under this map by and estimate what proportion of regular semisimple, semisimple and regular elements of it contains. We prove that as , all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups and .

Paper Structure

This paper contains 6 sections, 17 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Finite groups of Lie type
  3. Asymptotics of the power map for finite groups of Lie type
  4. The asymptotic results for powers in $\textup{GL}(n,q)$
  5. The asymptotic results for powers in $\textup{U}(n,q)$
  6. Some Examples

Key Result

Theorem 1.1

Let $G$ be a connected reductive group defined over $\mathbb F_q$ with Frobenius map $F$. Let $M\geq 2$ be an integer. Then, where the sum varies over non-conjugate maximal tori $T$ in $G(\mathbb F_q)$, $T=T_{d_1,\cdots, d_s}\cong C_{d_1}\times \cdots \times C_{d_s}$ reflects the cyclic structure of $T$, and the group $W_{T}=N_{G(\mathbb F_q)}(T)/T$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 27 more