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 AG (n, q). This generalizes research on Cameron-Liebler k-sets in the projective geometry PG(n, q). Note that in algebraic combinatorics, Cameron-Liebler k-sets of AG(n, q) correspond to certain equitable bipartitions of the association scheme of k-spaces in AG(n, q), while in the analysis of Boolean functions, they correspond to Boolean degree 1 functions of AG(n, q).We define Cameron-Liebler k-sets in AG(n, q) by intersection properties with kspreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler k-sets in AG(n, q) and PG(n, q). As a byproduct, 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 AG(n, q) for n > 3, while the case for Cameron-Liebler line classes in AG(3, q) was already treated separately.