Spectral Algebras of Abelian Cayley Graphs

Let $G$ be a finite abelian group of order $N$, $S \subseteq G \setminus \{0\}$ a symmetric connection set, and $K$ a field with $char(K) \nmid N$. The spectral algebra $\mathscr{A_K}(Cay(G,S)) = K[A]$ generated by the adjacency matrix of the Cayley graph is proved to decompose, via the character-or