Metadata
bibkey: cuntz2009integralmodulardata
authors: Michael Cuntz
year: 2009
title: Integral modular data and congruences
doi: 10.1007/s10801-008-0139-y
url: https://arxiv.org/abs/math/0611233v2
claim: Section 7(5) asks whether an integral Fourier matrix has a zero-free column containing an entry other than 1 or −1.
strata_touched:
- D5/S3/Fourier/CuntzZeroFreeColumnRefutation
license: citation-only
triage: anchor
Verified locator
DOI: 10.1007/s10801-008-0139-y
Source: https://arxiv.org/abs/math/0611233v2
Source statement
Section 7, Question (5), arXiv v2 p. 21, journal p. 380:
Does an integral Fourier matrix s exist with a column k such that ∀i sᵢₖ ≠ 0 and ∃i sᵢₖ ≠ ±1?
Definition 4.1 (arXiv v2 pp. 8–9) starts:
Let s be an integral Fourier matrix, i.e. s ∈ ℤ^{n×n} and
Its conditions (1)–(4) are the first column sᵢ₁ = 1 and det(s) ≠ 0; s sᵀ = diag(d₁,…,dₙ); √dⱼ sᵢⱼ = √dᵢ sⱼᵢ; and the integrality of every sum ∑ₖ sₖᵢ sₖⱼ sₖₗ/dₖ. Definition 3.1 (arXiv v2 pp. 3–4) states:
Here, √dᵢ is chosen greater than 0.
Definition 4.1 calls these matrices integral Fourier matrices before imposing the extra congruence conditions. No T-matrix is required for Question (5). The defining matrix equation determines row norms; the reversed indices in the explanatory norm sum of Definition 3.1 do not add another equation.
Scope
The existential question has a negative answer by a weighted column-square identity and integer-square rigidity. This does not classify integral Fourier matrices, decide the equal-norm question, or establish the existence of a T-matrix. Completeness of the external literature screen is ASSUMED-UNVERIFIED; the specific scope and unread sources are recorded in issue #22614.