{
  "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 $λ_χ= \\sum_{s \\in S} χ(s)$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09222",
  "author_total": 3,
  "authors": [
    {
      "name": "Deep Bhattacharjee"
    },
    {
      "name": "Priyabrata Mandal"
    },
    {
      "name": "Ushashi Bhattacharya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8qdbaqhy65vqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09222",
  "fulltext_url": null,
  "id": "8qdbaqhy65vqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09222",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212119184",
  "pdf_url": "https://arxiv.org/pdf/2609.09222",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8qdbaqhy65vqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09222",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Spectral Algebras of Abelian Cayley Graphs",
  "title_text": "Spectral Algebras of Abelian Cayley Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/8qdbaqhy65vqf",
  "venue": "arXiv",
  "versions": []
}