发表机构
Electro-Gravitational Space Propulsion Laboratory (EGSPL); Department of Mathematics, Bioinformatics and Computer Applications, Maulana Azad National Institute of Technology Bhopal; National Taiwan University（电引力空间推进实验室; 莫拉纳阿扎德国立技术学院博帕尔分校数学、生物信息学和计算机应用系; 国立台湾大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限阿贝尔群凯莱图的谱代数，通过特征标轨道分解将其分解为域扩张的半单乘积，并给出维数、幂等元及显式本原幂等元计算，涵盖笛卡尔积、初等阿贝尔群及特征p情形。
AI 中文摘要
设$G$为阶为$N$的有限阿贝尔群，$S \subseteq G \setminus \{0\}$为对称连接集，$K$为满足$char(K) \nmid N$的域。本文证明了由凯莱图的邻接矩阵生成的谱代数$\mathscr{A_K}(Cay(G,S)) = K[A]$，通过特征标轨道分解，可分解为$K$的域扩张的半单乘积，每个因子对应于特征值$\lambda_\chi = \sum_{s \in S} \chi(s)$的一个$Gal(\overline{K}/K)$-轨道。证明利用了阿贝尔离散傅里叶变换将$A$对角化，利用伽罗瓦群在特征标群$\widehat{G}$上的作用将特征值划分为轨道，并利用中国剩余定理将无平方因子的最小多项式转化为Wedderburn乘积。$\mathscr{A_K}(Cay(G,S))$的维数等于不同特征值的个数，幂等元个数为$2^r$，其中$r$为轨道数，本原幂等元通过$K[x]$中的Bezout算法显式计算。在$\mathbb Q$上，每个Wedderburn和项都是分圆域$\mathbb Q(\zeta_N)$的实子域。新结果包括：凯莱图笛卡尔积的张量积比较；初等阿贝尔群$(\mathbb Z/p)^k$的谱代数的系统分析（当$p \le 3$时为有理数，当$p \ge 5$时需要实分圆扩张）；以及非循环群（包括$\mathbb Z/6 \times \mathbb Z/2$和$\mathbb Z/5 \times \mathbb Z/2$）的详细轨道分析。循环情形恢复了伴随结果$\mathscr{A}_{\mathbb{Q}}(C_n) \cong \prod_{d \mid n} \mathbb Q(\zeta_d)^+$；汉明立方体给出$\mathscr{A}_\mathbb{Q}(\mathbb Q_k) \cong \mathbb{Q}^{k+1}$。本文还处理了特征为$p$的情形。
英文摘要
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 $λ_χ= \sum_{s \in S} χ(s)$. 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 $\mathbb Q(ζ_N)$. 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 $\mathscr{A}_{\mathbb{Q}}(C_n) \cong \prod_{d \mid n} \mathbb Q(ζ_d)^+$; the Hamming cube gives $\mathscr{A}_\mathbb{Q}(\mathbb Q_k) \cong \mathbb{Q}^{k+1}$. The characteristic-$p$ case is also treated.
