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Huixuan He

Publications and source records attributed to Huixuan He.

3 recordsLinked to original sources

arXiv2026

Coxeter Rigidity and ADE Classification of Fibonacci-Generated Tensor Categories

We propose a program to classify tensor categories generated by prescribed seed tensor categories. We carry out this program for finitely many marked copies of the positive Fibonacci category with pairwise nonisomorphic generating simple objects, in the setting of split locally finite semisimple rigid $\CC$-linear tensor categories with simple unit. We show that the Fibonacci fusion relation forces the fusion rules to be those of a specialized Coxeter--Hecke ring. The systems admitting categorification have Coxeter exponents $2,3,\infty$. For every such system we construct a tensor category and prove uniqueness up to marked tensor equivalence. Its simple objects are indexed by the Coxeter group, which may be infinite. Finite rank occurs precisely for disjoint unions of finite ADE diagrams, giving the ADE classification of Fibonacci-generated fusion categories. These categories are unitarizable, with no unitary structure assumed on the candidate. In finite ADE type, allowing Yang--Lee seeds gives one independent tensor-branch choice on each connected component.

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arXiv2026

Finite Composites of the Fibonacci Category with Pointed Fusion Categories

We classify finite composites of the positive Fibonacci category with arbitrary pointed fusion categories, retaining the entire source functor and making no unitarity assumption on the target. Formal codegrees, dimension fields, and algebraic integrality exclude all dihedral Fibonacci rings $R_n$ with $n\ge4$ from categorification. Together with an independent graphical uniqueness argument, this gives the finite two-Fibonacci classification. For several marked Fibonacci generators, a geometric height-and-polygon argument identifies the based ring with a standard-basis Coxeter--Hecke ring. Transitive conjugation by the pointed image forces the noncommutation graph to be empty or a perfect matching. We then obtain a crossed-product classification in terms of a quotient group, a stabilizer, labelled Fibonacci or rank-six $\TT$ blocks, and compatible cocycle and categorical action data. This includes the Fibonacci--$\Z/3\Z$ composite problem without target unitarity. We also establish the corresponding forms and marked descent statements over characteristic-zero fields admitting the chosen Fibonacci-type input.

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arXiv2026

Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture

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.

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