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

Abstract

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.