紧群的分段线性等变映射
Piecewise Linear Equivariant Maps for Compact Groups
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中文总结 AI 辅助
本文研究紧群有限维实表示间的分段线性等变映射，揭示其非线性行为的局限区域，得到紧群版本的Gibson-Tubbenhauer-Williamson存在准则，发现单位分支带来有限群无的刚性现象。
中文摘要 AI 辅助
受等变神经网络的启发，我们研究紧群的有限维实表示之间的分段线性等变映射。我们证明，所有真正的非线性分段线性行为都局限于群的单位分支平凡作用的子空间上，而等变性迫使在对应的正交补空间上呈线性。由此，我们得到了紧群版本的Gibson-Tubbenhauer-Williamson准则，该准则用于判断不可约表示之间非零等变分段线性映射的存在性；其中单位分支产生了有限群情形中不存在的刚性现象。
英文摘要
Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.
发表机构
- Università di Roma Sapienza（罗马智慧大学）
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