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 ${\mathcal{A}}_{\mathcal{K}}(Cay(G,S))=K[A]$ 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 $\hat{G}$ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of ${\mathcal{A}}_{\mathcal{K}}(Cay(G,S))$ 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 ${\mathcal{A}}_{\mathbb{Q}}(C_n)\cong {\prod }_{d\mid n}\mathbb{Q}({\zeta }_d{)}^{+}$; the Hamming cube gives ${\mathcal{A}}_{\mathbb{Q}}({\mathbb{Q}}_k)\cong {\mathbb{Q}}^{k+1}$. The characteristic-$p$ case is also treated.