Fourier Decomposition and E-Eigenvalues of Hypergraph Polynomials over Finite Abelian Groups
Abstract
Let H=(V,E) be a k-uniform hypergraph with polynomial fH(x)=∑e∈ E∏v∈ exv. For a finite abelian group G and a subgroup S of G, the k-partite Cayley hypergraph K has k disjoint vertex parts, each indexed by G, and {(1,g1),…,(k,gk)} is an edge if and only if g1+⋯+gk∈ S. For k=2, K is isomorphic to the Bi-Cayley graph associated with G and S. For k≥3, we use character theory to express fK, under a complex unitary change of variables, as a sum of exactly [G:S] nonzero monomials in pairwise disjoint sets of variables. This yields a direct sum decomposition whenever S is a proper subgroup of G. When G=Z2m, this transformation is real and orthogonal, enabling us to determine the complete sets of complex E-eigenvalues and Z-eigenvalues of the adjacency tensor of K.
Related
- On Intersection Hypergraph of Finite Non-Cyclic Abelian Groups
- Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups
- General Polynomials and Eigenvalues Over Cayley--Dickson Algebras
- Ehrhart quasi-polynomials with finite abelian congruence constraints
- Divergence of decreasing rearranged Fourier series on every infinite compact abelian group
BibTeX
@misc{bao2026fourier,
title = {{Fourier Decomposition and E-Eigenvalues of Hypergraph Polynomials over Finite Abelian Groups}},
author = {Yan-Hong Bao and Yi-Zheng Fan and Lili Liao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28177},
eprint = {2610.28177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28177},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fourier Decomposition and E-Eigenvalues of Hypergraph Polynomials over Finite Abelian Groups
AU - Yan-Hong Bao
AU - Yi-Zheng Fan
AU - Lili Liao
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28177
UR - https://arxiv.org/abs/2610.28177
AB - Let H=(V,E) be a k-uniform hypergraph with polynomial f_H(x)=∑_e∈ E∏_v∈ ex_v. For a finite abelian group G and a subgroup S of G, the k-partite Cayley hypergraph K has k disjoint vertex parts, each indexed by G, and {(1,g_1),…,(k,g_k)} is an edge if and only if g_1+⋯+g_k∈ S. For k=2, K is isomorphic to the Bi-Cayley graph associated with G and S. For k≥3, we use character theory to express f_K, under a complex unitary change of variables, as a sum of exactly [G:S] nonzero monomials in pairwise disjoint sets of variables. This yields a direct sum decomposition whenever S is a proper subgroup of G. When G=Z_2^m, this transformation is real and orthogonal, enabling us to determine the complete sets of complex E-eigenvalues and Z-eigenvalues of the adjacency tensor of K.
ER -