Global Symmetries, Local Symmetries and Groupoids

Abstract

Local symmetries are primarily defined in the case of spacetime, but several authors have defined them outside this context, sometimes with the help of groupoids. We show that, in many cases, local symmetries can be defined as global symmetries. We also show that groups can be used, rather than groupoids, to handle local symmetries. Examples are given for graphs and networks, color symmetry and tilings. The definition of local symmetry in physics is also discussed.

Eigen-Persistence in Graphs

“…Some attention has been directed to the occurrence of excessive (eigenvalue) degeneracies [17][18][19][20][21][22][23], sometimes viewed as manifestations of "local" structures [24][25][26][27][28][29][30][31][32]. Since degeneracies arose early on in the context of symmetry (and group theory), there has been a natural effort [33][34][35][36][37][38][39] to investigate excessive degeneracies in terms of symmetries, most simply in terms of a graph's automorphism group, which generally goes beyond standard point-group symmetries. The much studied [40][41][42][43][44][45] eigen-solution to the "Bethe tree" manifests high degeneracies.…”

Chirality in Affine Spaces and in Spacetime

“…It may be global or local. The potential existence of chiral objects is also a property of the space, which may also stand globally or locally [11]. Both properties depend on the space structure, but none of them is the cause of the existence of the other one.…”