主单元Diţă型Butson矩阵：特征标表变换、分式线性自同构、Weyl阈值与极化Heisenberg窗口

中文摘要 AI 辅助

对于奇素数p和整数r,q≥1，我们研究在ℤ/p^qℤ上的Butson型复Hadamard矩阵H_{p;r,q}(a,b)=χ(1+p^r ab)，其中χ是主单元商群的忠实特征。定义相位为归一化p进对数，当q≤r时模p^q线性，当q>r时非线性。我们证明该阈值同时控制与有限阿贝尔特征标表的单项等价性、典范循环2-上链的上闭链性质，以及典范Weyl换位子的投射标量性。我们将全投射单项自同构群分类为阶为p^{2q}φ(p^q)的三参数分式线性群，其逆极限为PGL₂(ℤₚ)的Γ₀(p^r)型同余子群。对于q>r，标量换位子仅在恰好q-r+1个极大赋值极化窗口上存在；商去根基后，每个窗口携带阶为p^r的两个循环群上的标准完美配对及相关有限Heisenberg扩张。我们确定这些窗口的正规化子，并证明不同极化在投射单项群内互不共轭。最后，对每个q>r，阿贝尔化将H_{p;r,q}与Fourier矩阵F_{p^q}区分；低阶缺陷与指纹计算说明该群论不变量在该族上为何更强。

英文摘要

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.

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19 pages