On complete morphisms of Butson Hadamard matrices

@misc{egan2026complete,
  title = {{On complete morphisms of Butson Hadamard matrices}},
  author = {Ronan Egan and Pádraig Ó Catháin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28394},
  eprint = {2610.28394},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28394},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On complete morphisms of Butson Hadamard matrices
AU  - Ronan Egan
AU  - Pádraig Ó Catháin
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28394
UR  - https://arxiv.org/abs/2610.28394
AB  - A complete morphism of Butson classes replaces each entry of a matrix in BH(n,k) by an m × m block, uniformly, to produce a matrix in BH(mn,t). We show that every complete morphism is described by a pair (U,Z), where Z ∈ BH(m,t) and U is a unitary matrix annihilated by the k^th cyclotomic polynomial. We deduce from a theorem of Lam and Leung that complete morphisms onto real Hadamard matrices exist precisely when k is a power of $2$. The exponent-reducing morphisms of Östergård--Paavola and Ó Catháin--Swartz achieve the smallest block size the degree bound allows, and we show that there are no other complete morphisms for which U is monomial. We construct new complete morphisms where U is not monomial: a complete morphism BH(n,7) → BH(6n,6) is given, the first to eliminate a prime from the alphabet.
ER  -