# Codes and Expansions (CodEx) Seminar

## Gary R. W. Greaves (Nanyang Technological University)

Hermitian matrices of roots of unity and their characteristic polynomials

Many known configurations of Equiangular Tight Frames correspond to Hermitian matrices of roots of unity. Motivated by the pursuit of necessary conditions for their existence, we consider spectral properties of such matrices. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of a Hermitian matrix of roots of unity modulo ideals generated by powers of \(1-\zeta\), where \(\zeta\) is a root of unity. During this talk, I will present 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.