Spectral Algebras of Abelian Cayley Graphs

Abstract

Let G be a finite abelian group of order N, S ⊆ G ∖ {0} a symmetric connection set, and K a field with char(K) nmid N. The spectral algebra 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(K̅/K)-orbit of the eigenvalues λ_χ= ∑_{s ∈ S} χ(s). The proof uses the abelian discrete Fourier transform to diagonalise A, the Galois action on the character group Ĝ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of 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 ℚ, every Wedderburn summand is a real subfield of the cyclotomic field 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 (ℤ/p)^k (rational for p ≤ 3, requiring real cyclotomic extensions for p ≥ 5); and a worked orbit analysis for non-cyclic groups including ℤ/6 × ℤ/2 and ℤ/5 × ℤ/2. The cyclic case recovers the companion result 𝒜_ℚ(C_n) ≅ ∏_{d | n} ℚ(ζ_d)⁺; the Hamming cube gives 𝒜Q(ℚ_k) ≅ ℚ^k+1. The characteristic-p case is also treated.

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Cite

BibTeX RIS

BibTeX

@misc{bhattacharjee2026spectral,
  title = {{Spectral Algebras of Abelian Cayley Graphs}},
  author = {Deep Bhattacharjee and Priyabrata Mandal and Ushashi Bhattacharya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09222},
  eprint = {2609.09222},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09222},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Spectral Algebras of Abelian Cayley Graphs
AU  - Deep Bhattacharjee
AU  - Priyabrata Mandal
AU  - Ushashi Bhattacharya
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.09222
UR  - https://arxiv.org/abs/2609.09222
AB  - Let G be a finite abelian group of order N, S ⊆ G ∖ {0} a symmetric connection set, and K a field with char(K) nmid N. The spectral algebra 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(K̅/K)-orbit of the eigenvalues λ_χ= ∑_s ∈ S χ(s). The proof uses the abelian discrete Fourier transform to diagonalise A, the Galois action on the character group Ĝ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of 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 ℚ, every Wedderburn summand is a real subfield of the cyclotomic field 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 (ℤ/p)^k (rational for p ≤ 3, requiring real cyclotomic extensions for p ≥ 5); and a worked orbit analysis for non-cyclic groups including ℤ/6 × ℤ/2 and ℤ/5 × ℤ/2. The cyclic case recovers the companion result 𝒜_ℚ(C_n) ≅ ∏_d | n ℚ(ζ_d)^+; the Hamming cube gives 𝒜_Q(ℚ_k) ≅ ℚ^k+1. The characteristic-p case is also treated.
ER  -

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