The higher Segal degree of a partial groupoid equals the Helly number of the closure space of a characteristic action, and the method gives explicit degrees for punctured Weyl groups.
Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins
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abstract
In this paper, we construct a partial group \(\mathcal{P}(F)\) that represents the "partial symmetry" inherent in a subset \(F\) of \(d\)-dimensional Euclidean space. In cases where \(F\) is not connected, \(\mathcal{P}(F)\) captures more detailed information than the conventional symmetry group \(G(F)\). To establish a stronger connection between \(\mathcal{P}(F)\) and \(F\), we introduce a novel definition of partial group action. Furthermore, to characterize \(\mathcal{P}(F)\) in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.
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2025 1verdicts
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Higher Segal spaces and partial groups
The higher Segal degree of a partial groupoid equals the Helly number of the closure space of a characteristic action, and the method gives explicit degrees for punctured Weyl groups.