Principal-unit Diţă-type Butson matrices: character-table transitions, fractional-linear automorphisms, Weyl thresholds, and polarized Heisenberg windows

Abstract

For an odd prime $p$ and integers $r,q\ge 1$, we study the Butson-type complex Hadamard matrices $H_{p;r,q}(a,b)=χ(1+p^r ab)$ on $\mathbb{Z}/p^q\mathbb{Z}$, where $χ$ is a faithful character of the principal-unit quotient. The defining phase is the normalized $p$-adic logarithm, linear modulo $p^q$ exactly when $q\le r$ and nonlinear when $q>r$. We prove that this threshold simultaneously governs monomial equivalence to a finite abelian character table, the cocycle property of the canonical cyclic $2$-cochain, and projective scalarity of the canonical Weyl commutator. We classify the full projective monomial automorphism group as a three-parameter fractional-linear group of order $p^{2q}φ(p^q)$, with inverse limit a $Γ_0(p^r)$-type congruence subgroup of $\mathrm{PGL}_2(\mathbb{Z}_p)$. For $q>r$, scalar commutators survive on exactly $q-r+1$ maximal valuation-polarized windows; after quotienting radicals, each carries the standard perfect pairing on two cyclic groups of order $p^r$ and the associated finite Heisenberg extension. We determine the exact normalizers of these windows and prove distinct polarizations are pairwise nonconjugate inside the projective monomial group. Finally, abelianization separates $H_{p;r,q}$ from the Fourier matrix $F_{p^q}$ for every $q>r$; low-order defect and fingerprint computations illustrate why this group-theoretic invariant is stronger on the family.