Hermitian matrices of roots of unity and their characteristic polynomials

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2023-11, Vol.200, p.105793, Article 105793
Main Authors: Greaves, Gary R.W., Woo, Chin Jian
Format: Article
Language:English
Subjects:
Characteristic polynomials
Equiangular tight frames
Hermitian matrices
Roots of unity
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Summary:We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2023.105793