发表机构
School of Mathematics, University of Birmingham（伯明翰大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明类型 B 和 C 的有限单群满足归纳分块 Alperin 权条件，通过置换格、Jordan 约化等方法去除单位三角假设，并补全散在群情形，从而推出相关群的块权猜想成立。
AI 中文摘要
本文旨在证明所有类型 $\mathsf B$ 或 $\mathsf C$ 的有限单群在其阶的每个素数 $\ell$ 上都满足归纳分块 Alperin 权条件。对于 ${\rm PSp}_{2n}(q)$，其中 $q$ 为奇数且 $n\geq3$，基于 Conlon 归纳定理的置换格论证去除了早期关于奇数非定义素数归纳条件结果中的单位三角假设。在奇数定义特征下 $\ell=2$ 时，底层 radical $2$-子群分类中的错误和遗漏影响了先前对任意块权参数的刻画。我们验证了主块的所需权参数，并使用 Jordan 约化建立了所有块的归纳条件。在偶数定义特征且秩至少为四时，我们在不假设单位三角性的情况下，利用一般权和 Jordan 约化证明了归纳条件。对于 $\Omega_{2n+1}(q)$，其中 $q$ 为奇数且 $n\geq3$，我们使用 Conlon 归纳定理去除了奇数非定义素数下的单位三角假设。在 $\ell=2$ 时，我们证明了 $\operatorname{Spin}_{2n+1}(q)$ 的 Brauer 字符的稳定子和扩张性质，该性质在早期关于归纳条件的工作中被假定。我们还给出了剩余散在单群 $J_4$、$Fi'_{24}$、大魔群和魔群的归纳条件的证明，且不主张优先权。结合先前已确立的情形，这些结果意味着：对于每个有限群，若其每个非交换单截面（其阶被 $\ell$ 整除）为类型 $\mathsf B$、类型 $\mathsf C$ 或散在单群，则分块 Alperin 权猜想在 $\ell$ 处成立。
英文摘要
The purpose of this paper is to prove that every finite simple group of type $\mathsf B$ or $\mathsf C$ satisfies the inductive blockwise Alperin weight condition at every prime $\ell$ dividing its order. For ${\rm PSp}_{2n}(q)$, where $q$ is odd and $n\geq3$, 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 $\ell=2$ 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 $\ell$ 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 $n\geq3$, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At $\ell=2$, we prove the stabiliser and extension property for Brauer characters of $\operatorname{Spin}_{2n+1}(q)$ 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 $\ell$ for every finite group each of whose nonabelian simple sections of order divisible by $\ell$ is of type $\mathsf B$, of type $\mathsf C$, or sporadic.
CommentsFor information on the Lean sources and reflections on the use of AI in mathematical research, please see the acknowledgements
