Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture
Abstract
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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BibTeX
@misc{he2026triangular,
title = {{Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture}},
author = {Huixuan He and Zhengwei Liu and Fan Lu and Sebastien Palcoux and Yunxiang Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11652},
eprint = {2610.11652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11652},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture
AU - Huixuan He
AU - Zhengwei Liu
AU - Fan Lu
AU - Sebastien Palcoux
AU - Yunxiang Ren
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11652
UR - https://arxiv.org/abs/2610.11652
AB - 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.
ER -