发表机构
Tsinghua University; Beijing Institute of Mathematical Sciences and Applications; LinkedIn Corporation（清华大学; 北京国际数学研究中心; 领英公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推导近群范畴的三角棱柱方程，利用双曲伽马函数构造循环群的TPE解，建立正维数标量解的酉性判据，揭示范畴酉性与希尔伯特第十二问题的联系。
AI 中文摘要
我们推导了类型为$G+|G|$的近群融合规则的三角棱柱方程（TPE），其中$G$为有限阿贝尔群。我们证明：在任意使$|G|$可逆的域上，具有平凡Frobenius-Schur示性标的球面范畴化等价于TPE方程组的可解性。对每个循环群，我们利用双曲伽马函数构造了TPE的显式解，得到了具有负单对象维数的球面范畴，且经伽罗瓦共轭后得到具有正维数的伪酉范畴。对具有正维数的标量解，我们建立了精确奇异值公式和一致间隙，为酉性提供了显式充分判据。最后，我们证明：任意阶循环伽罗瓦共轭系数的酉性对应于一阶阿贝尔Stark猜想所隐含的实乘法猜想的特例，揭示了范畴酉性与实二次域的希尔伯特第十二问题之间的深刻联系。
英文摘要
We derive the triangular prism equations (TPE) for near-group fusion rules of type $G+|G|$, where $G$ is a finite abelian group. We prove that spherical categorification with trivial Frobenius--Schur indicator over any field in which $|G|$ is invertible is equivalent to the solvability of the TPE system. For every cyclic group, we construct an explicit solution to the TPE system using hyperbolic gamma functions, yielding both a spherical category with negative simple-object dimensions and, after Galois conjugation, a pseudo-unitary category with positive dimensions. For scalar solutions with positive dimensions, we establish an exact singular-value formula and a uniform gap, which provide explicit sufficient criteria for unitarity. Finally, we show that the unitarity of the cyclic Galois-conjugated coefficients for arbitrary orders corresponds to a specialization of a real-multiplication conjecture implied by the order-one abelian Stark conjecture, revealing a deep connection between categorical unitarity and Hilbert's twelfth problem for real quadratic fields.
Comments40 pages, 4 figures
