Cameron-Liebler \(k\)-sets in \(\text{AG}(n,q)\)

We study Cameron-Liebler \(k\)-sets in the affine geometry, so sets of \(k\)-spaces in \(\text{AG}(n, q)\). This generalizes research on Cameron-Liebler \(k\)-sets in the projective geometry \(\text{PG}(n, q)\). Note that in algebraic combinatorics, Cameron-Liebler \(k\)-sets of \(\text{AG}(n, q)\)...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-03
Main Authors: D'haeseleer, Jozefien, Ihringer, Ferdinand, Mannaert, Jonathan, Storme, Leo
Format: Article
Language:English
Subjects:
Boolean
Boolean algebra
Boolean functions
Combinatorial analysis
Projective geometry
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study Cameron-Liebler \(k\)-sets in the affine geometry, so sets of \(k\)-spaces in \(\text{AG}(n, q)\). This generalizes research on Cameron-Liebler \(k\)-sets in the projective geometry \(\text{PG}(n, q)\). Note that in algebraic combinatorics, Cameron-Liebler \(k\)-sets of \(\text{AG}(n, q)\) correspond to certain equitable bipartitions of the Association scheme of \(k\)-spaces in \(\text{AG}(n, q)\), while in the analysis of Boolean functions, they correspond to Boolean degree \(1\) functions of \(\text{AG}(n, q)\). We define Cameron-Liebler \(k\)-sets in \(\text{AG}(n, q)\) by intersection properties with \(k\)-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler \(k\)-sets in \(\text{AG}(n, q)\) and \(\text{PG}(n, q)\). As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler \(k\)-sets. This paper focuses on \(\text{AG}(n, q)\) for \(n > 3\), while the case for Cameron-Liebler line classes in \(\text{AG}(3, q)\) was already treated separately.
ISSN:2331-8422
DOI:10.48550/arxiv.2003.12429