Spectral Algebras of Abelian Cayley Graphs

Abstract

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-orbit decomposition, as a semisimple product of field extensions of $K$, one factor for each $Gal(\overline{K}/K)$-orbit of the eigenvalues $\lambda_\chi = \sum_{s \in S} \chi(s)$. The proof uses the abelian discrete Fourier transform to diagonalise $A$, the Galois action on the character group $\widehat{G}$ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of $\mathscr{A_K}(Cay(G,S))$ equals the number of distinct eigenvalues, the idempotent count is $2^r$ where $r$ is the orbit number, and primitive idempotents are computed explicitly via the Bezout algorithm in $K[x]$. Over $\mathbb Q$, every Wedderburn summand is a real subfield of the cyclotomic field $\mathbb Q(\zeta_N)$. New results include: a tensor-product comparison for Cartesian products of Cayley graphs; a systematic analysis of the spectral algebra for elementary abelian groups $(\mathbb Z/p)^k$ (rational for $p \le 3$, requiring real cyclotomic extensions for $p \ge 5$); and a worked orbit analysis for non-cyclic groups including $\mathbb Z/6 \times \mathbb Z/2$ and $\mathbb Z/5 \times \mathbb Z/2$. The cyclic case recovers the companion result $\mathscr{A}_{\mathbb{Q}}(C_n) \cong \prod_{d \mid n} \mathbb Q(\zeta_d)^+$; the Hamming cube gives $\mathscr{A}_\mathbb{Q}(\mathbb Q_k) \cong \mathbb{Q}^{k+1}$. The characteristic-$p$ case is also treated.