Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

2025/06/17 by Takahiro Hayashi, Hayashi, Takahiro
Materials Science · #08A99 #18N50 #51M20 #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2506.14304

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

In this paper, we construct a partial group \(P(F)\) that represents the "partial symmetry" inherent in a subset \(F\) of \(d\)-dimensional Euclidean space. In cases where \(F\) is not connected, \(P(F)\) captures more detailed information than the conventional symmetry group \(G(F)\). To establish a stronger connection between \(P(F)\) and \(F\), we introduce a novel definition of partial group action. Furthermore, to characterize \(P(F)\) in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.

Citations

• Higher Segal spaces and partial groups
• Simplicial effects and weakly associative partial groups
• Partial groups as symmetric simplicial sets
• Limits and colimits, generators and relations of partial groups
• An extension theory for partial groups
• Semidirect Products and Wreath Products of Localities
• Finite Localities III
• An extension theory for partial groups and localities
• Finite Localities I
• Fusion systems and localities

Related

• Semidirect Products and Wreath Products of Localities
• Semigroup and Category-Theoretic Approaches to Partial Symmetry
• Partial actions of groups on monoidal categories
• Partial dynamical systems and C*-algebras generated by partial isometries
• From endomorphisms to automorphisms and back: dilations and full corners