有效的Bialynicki-Birula-Brosnan动机分解
Effective Bialynicki-Birula-Brosnan motivic decompositions
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中文总结 AI 辅助
本文针对内型等径约化群对应的射影齐性簇，实现了Brosnan提出的动机分解的SageMath算法，将其推广到特征2的经典群并给出例外群的新分解。
中文摘要 AI 辅助
设G为一个等径约化群，X为一个射影G齐性簇。利用Bialynicki-Birula、Hesselink及Iversen的结果，Brosnan证明：若G为内型，则X的动机可表示为G的非异核对应的射影齐性簇的塔特扭转动机的直和。我们提供该分解的SageMath实现，其基于G的外尔群的凯莱图的深度优先搜索，确保复杂度随动机的规模而非整个外尔群的规模变化。作为应用，我们给出若干例外群的新动机分解，并展示如何将Karpenko的经典群分解推广到特征2的情形。
英文摘要
Let $G$ be an isotropic reductive group and $X$ be a projective $G$-homogeneous variety. Using results from Bialynicki-Birula, Hesselink and Iversen, Brosnan showed that if $G$ is of inner type, the motive of $X$ can be expressed as a direct sum of Tate twists of motives of projective homogeneous varieties for the anisotropic kernel of $G$. We provide a SageMath implementation of this decomposition, based on a depth-first search of the Cayley graph of the Weyl group of $G$, ensuring the complexity scales with the size of the motive rather than the full Weyl group. As applications, we provide new motivic decompositions for some exceptional groups and show how to extend Karpenko's decompositions for classical groups to characteristic $2$.
