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

Let $H=(V,E)$ be a $k$-uniform hypergraph with polynomial $f_H(\mathit{x})={\sum }_{e\in E}{\prod }_{v\in e}{x}_v$. 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 $\{(1,g_1),\dots ,(k,g_k)\}$ is an edge if and only if $g_1+\cdots +g_k\in S$. For $k=2$, $K$ is isomorphic to the Bi-Cayley graph associated with $G$ and $S$. For $k\ge 3$, we use character theory to express $f_K$, under a complex unitary change of variables, as a sum of exactly $[G:S]$ nonzero monomials in pairwise disjoint sets of variables. This yields a direct sum decomposition whenever $S$ is a proper subgroup of $G$. When $G={\mathbb{Z}}_2^m$, this transformation is real and orthogonal, enabling us to determine the complete sets of complex E-eigenvalues and Z-eigenvalues of the adjacency tensor of $K$.