Morphisms of Butson classes

Abstract

We introduce the concept of a morphism from the set of Butson Hadamard matrices over k th roots of unity to the set of Butson matrices over ℓ th roots of unity. As concrete examples of such morphisms, we describe tensor-product-like maps which reduce the order of the roots of unity appearing in a Butson matrix at the cost of increasing the dimension. Such maps can be constructed from Butson matrices with eigenvalues satisfying certain natural conditions. Our work unifies and generalises Turyn's construction of real Hadamard matrices from Butson matrices over the 4 th roots and the work of Compton, Craigen and de Launey on 'unreal' Butson matrices over the 6 th roots. As a case study, we classify all morphisms from the set of n × n Butson matrices over k th roots of unity to the set of 2n × 2n Butson matrices over ℓ th roots of unity where ℓ <

Quasi-Hadamard Full Propelinear Codes

“…For one, a BH(n, k) exists for all n, (the Fourier matrix for example), but real Hadamard matrices, i.e., BH(n, 2), exist when n > 2 only if n ≡ 0 mod 4, and this condition is famously not yet known to be sufficient. A Butson morphism [8] is a map BH(n, k) → BH(m, ℓ). This motives the study of Butson matrices even if real Hadamard matrices are the primary interest.…”

“…For t 1 = 1, t 2 = 1, t 3 = 0 then L(H) is the matrix given in Example 1.1. For t 1 = 1, t 2 = 1, t 3 = 1 then H ∈ BH (8,8) where…”

“…x ∈ C 1 (1,2,3,4,5,6,7,8) x ∈ C 2 (1, 3, 5, 7) (2,4,6,8) x ∈ C 3 (1,4,7,2,5,8,3,6) x ∈ C 4 (1, 5)(2, 6)(3, 7)(4, 8)…”

Homomorphisms of matrix algebras and constructions of Butson–Hadamard​ matrices

“…More recently, Compton, Craigen and de Launey proved that a matrix in BH(n, 6) with no entries in {1, −1} implies that BH(4n, 2) is nonempty [3]. The first author and Egan unified and generalised these results, giving sufficient conditions for the existence of a matrix in BH(n, k) and a matrix in M ∈ BH(m, ℓ) to imply the existence of a matrix in BH(mn, ℓ) [4]. The most substantial conditions in these constructions are on the spectrum of the matrix M. In [5], the authors of this paper and Egan proved the existence of a real Hadamard matrix with minimal polynomial Φ 2 t+1 (x), which implies that, whenever there exists H ∈ BH(n, 2 t ), there exists a real Hadamard matrix of order 2 2 t−1 −1 n.…”

Spectra of Hadamard matrices

“…In [6], the authors define a (complete) morphism of Butson matrices to be a function from BH(n, k) to BH(r, ℓ). A partial morphism is a morphism such that the domain is a proper subset of BH(n, k).…”