A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum
Abstract
We study the rank-twelve potential modular datum denoted 1239,4064,1276. in the low-rank classification tables. Its global dimension is (1287+351√(13))/2, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on Hom(C[10],B⊗3). Generalized Frobenius--Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in B3 then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap 2/117 is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.
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BibTeX
@misc{liu2026multiplicative,
title = {{A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum}},
author = {Jiuhe Liu and Xuzhihan Ding},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15792},
eprint = {2610.15792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15792},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum
AU - Jiuhe Liu
AU - Xuzhihan Ding
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15792
UR - https://arxiv.org/abs/2610.15792
AB - We study the rank-twelve potential modular datum denoted 12^39,406_4,1276. in the low-rank classification tables. Its global dimension is (1287+351√(13))/2, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on Hom(C_[10],B^⊗3). Generalized Frobenius--Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in B_3 then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap 2/117 is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.
ER -