Fourier Decomposition and E-Eigenvalues of Hypergraph Polynomials over Finite Abelian Groups

Let $H=(V,E)$ be a $k$-uniform hypergraph with polynomial $f_H(\bm{x})=\sum_{e\in E}\prod_{v\in e}x_v$. For a finite abelian group $G$ and a subgroup $S$ of $G$, the $k$-partite Cayley hypergraph $K$ has $k$ disjoint vertex parts, each indexed by $G$, and $\{(1,g_1),\ldots,(k,g_k)\}$ is an edge if a