PreprintCC BY 4.0arXiv
The dimension block of a spherical fusion category
Abstract
Let 𝒞 be a spherical fusion category with global dimension D. The Galois conjugates of the dimension character index a block of the S-matrix of the Drinfeld center, which we call the dimension block. We prove that for every character χ of Gal(ℚ(D)/ℚ), the twisted sum ∑σχ(σ)/σ(D) has absolute value at most e−1f(χ)−1/2, where f(χ) is the conductor of χ and e is the order of the dimensional grading group of 𝒞. As an application, we determine the possible global dimensions of spherical fusion categories below √5, extending the known classification up to 4√3/5 of V.\ Ostrik and P.\ Etingof.
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BibTeX
@misc{schopieray2026dimension,
title = {{The dimension block of a spherical fusion category}},
author = {Andrew Schopieray},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35287},
eprint = {2609.35287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35287},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The dimension block of a spherical fusion category
AU - Andrew Schopieray
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35287
UR - https://arxiv.org/abs/2609.35287
AB - Let 𝒞 be a spherical fusion category with global dimension D. The Galois conjugates of the dimension character index a block of the S-matrix of the Drinfeld center, which we call the dimension block. We prove that for every character χ of Gal(ℚ(D)/ℚ), the twisted sum ∑_σχ(σ)/σ(D) has absolute value at most e^−1f(χ)^−1/2, where f(χ) is the conductor of χ and e is the order of the dimensional grading group of 𝒞. As an application, we determine the possible global dimensions of spherical fusion categories below √5, extending the known classification up to 4√3/5 of V.\ Ostrik and P.\ Etingof.
ER -