Finite Composites of the Fibonacci Category with Pointed Fusion Categories
Abstract
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 Rn with n≥4 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/3Z 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.
Related
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BibTeX
@misc{he2026finite,
title = {{Finite Composites of the Fibonacci Category with Pointed Fusion Categories}},
author = {Huixuan He and Zhengwei Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41891},
eprint = {2610.41891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41891},
}
RIS
TY - UNPB
TI - Finite Composites of the Fibonacci Category with Pointed Fusion Categories
AU - Huixuan He
AU - Zhengwei Liu
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41891
UR - https://arxiv.org/abs/2610.41891
AB - 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≥4 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/3Z 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.
ER -