On the inductive blockwise Alperin weight condition for type B and type C
Abstract
The purpose of this paper is to prove that every finite simple group of type B or C satisfies the inductive blockwise Alperin weight condition at every prime ℓ dividing its order. For PSp2n(q), where q is odd and n≥3, a permutation lattice argument based on Conlon's induction theorem removes the unitriangularity assumption from earlier results on the inductive condition at odd nondefining primes. At ℓ=2 in odd defining characteristic, errors and omissions in the underlying classification of radical $2$-subgroups affect an earlier parametrisation of weights for arbitrary blocks. We verify the required weight parametrisation for principal blocks and use Jordan reduction to establish the inductive condition for all blocks. At odd ℓ in even defining characteristic and rank at least four, we prove the inductive condition using generic weights and Jordan reduction, without assuming unitriangularity. For Ω2n+1(q), where q is odd and n≥3, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At ℓ=2, we prove the stabiliser and extension property for Brauer characters of Spin2n+1(q) that was assumed in earlier work on the inductive condition. We also give proofs of the inductive condition for the remaining sporadic groups J4, Fi'24, the Baby Monster and the Monster, without making a priority claim. Together with the previously established cases, these results imply that the blockwise Alperin weight conjecture holds at ℓ for every finite group each of whose nonabelian simple sections of order divisible by ℓ is of type B, of type C, or sporadic.
Related
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BibTeX
@misc{zhang2026inductive,
title = {{On the inductive blockwise Alperin weight condition for type $\mathsf B$ and type $\mathsf C$}},
author = {Baoyu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24774},
eprint = {2609.24774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24774},
}
RIS
TY - UNPB
TI - On the inductive blockwise Alperin weight condition for type B and type C
AU - Baoyu Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24774
UR - https://arxiv.org/abs/2609.24774
AB - The purpose of this paper is to prove that every finite simple group of type B or C satisfies the inductive blockwise Alperin weight condition at every prime ℓ dividing its order. For PSp_2n(q), where q is odd and n≥3, a permutation lattice argument based on Conlon's induction theorem removes the unitriangularity assumption from earlier results on the inductive condition at odd nondefining primes. At ℓ=2 in odd defining characteristic, errors and omissions in the underlying classification of radical $2$-subgroups affect an earlier parametrisation of weights for arbitrary blocks. We verify the required weight parametrisation for principal blocks and use Jordan reduction to establish the inductive condition for all blocks. At odd ℓ in even defining characteristic and rank at least four, we prove the inductive condition using generic weights and Jordan reduction, without assuming unitriangularity. For Ω_2n+1(q), where q is odd and n≥3, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At ℓ=2, we prove the stabiliser and extension property for Brauer characters of Spin_2n+1(q) that was assumed in earlier work on the inductive condition. We also give proofs of the inductive condition for the remaining sporadic groups J_4, Fi'_24, the Baby Monster and the Monster, without making a priority claim. Together with the previously established cases, these results imply that the blockwise Alperin weight conjecture holds at ℓ for every finite group each of whose nonabelian simple sections of order divisible by ℓ is of type B, of type C, or sporadic.
ER -