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Cameron-Liebler sets for maximal totally isotropic flats in classical affine spaces

Jun Guo, Lingyu Wan

Abstract

Let $ACG(2ν,\mathbb{F}_q)$ be the $2ν$-dimensional classical affine space with parameter $e$ over a $q$-element finite field $\mathbb{F}_q$, and ${\cal O}_ν$ be the set of all maximal totally isotropic flats in $ACG(2ν,\mathbb{F}_q)$. In this paper, we discuss Cameron-Liebler sets in ${\cal O}_ν$, obtain several equivalent definitions and present some classification results.

Cameron-Liebler sets for maximal totally isotropic flats in classical affine spaces

Abstract

Let be the -dimensional classical affine space with parameter over a -element finite field , and be the set of all maximal totally isotropic flats in . In this paper, we discuss Cameron-Liebler sets in , obtain several equivalent definitions and present some classification results.
Paper Structure (6 sections, 28 theorems, 84 equations, 1 table)

This paper contains 6 sections, 28 theorems, 84 equations, 1 table.

Table of Contents

  1. Introduction
  2. Classical affine spaces
  3. Association schemes
  4. Equivalent definitions
  5. Classification results
  6. Concluding remarks

Key Result

Lemma 2.1

(See GruenbergWan). Let $F_1=V_1+x_1$ and $F_2=V_2+x_2$ be any two flats in $ACG(2\nu,\mathbb{F}_q)$, where $V_1$ and $V_2$ are two subspaces of $\mathbb{F}_q^{2\nu}$, and $x_1,x_2\in\mathbb{F}_q^{2\nu}$. Then the following hold.

Theorems & Definitions (30)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 20 more