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

Abstract

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$.
 We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ 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 $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. 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 $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

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

“…Similar problems have been investigated by various researchers under different names: Boolean degree one functions, completely regular codes of strength 0 and covering radius 1, and tight sets, see [19] for more details on these connections. Recently, De Boeck et al [13, 11] studied Cameron–Liebler sets of generators in polar spaces, and D'haeseleer et al [15, 16] studied Cameron–Liebler sets in $$AG(n,q)$$. Their research stimulate us to consider Cameron–Liebler sets in classical affine spaces.…”

Approximately strongly regular graphs

“…In recent years, regular sets in graphs coming from finite geometries have received a significant amount of attention, for instance see [2,3,4,10,12,13,20,21,22,25,27,28,29,36,37,38]. Note that the investigation of regular sets in conjugacy class schemes is also of recent interest, for instance see [17,19]. Often these association schemes are distance-regular graphs.…”