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Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

Filed under: General — Tags: , — m759 @ 2:34 am

Last revised: January 20, 2023 @ 11:39:05

The First Approach — Via Substructure Isomorphisms —

From "Symmetry in Mathematics and Mathematics of Symmetry"
by Peter J. Cameron, a Jan. 16, 2007, talk at the International
Symmetry Conference, Edinburgh, Jan. 14-17, 2007

Local or global?

"Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M  is homogeneous * if every isomorphism between finite substructures of M  can be extended to an automorphism of ; in other words, 'any local symmetry is global.' "

A related discussion of the same approach — 

"The aim of this thesis is to classify certain structures
which are, from a certain point of view,
as homogeneous as possible, that is
which have as many symmetries as possible.
the basic idea is the following: a structure S  is
said to be homogeneous  if, whenever two (finite)
substructures Sand S2 of S  are isomorphic,
there is an automorphism of S  mapping S1 onto S2.”

— Alice Devillers,
Classification of Some Homogeneous
and Ultrahomogeneous Structures
,”
Ph.D. thesis, Université Libre de Bruxelles,
academic year 2001-2002

The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.

For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:

Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.

Related material:

Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2

and, more recently, 

Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)

This approach seems to be a rather deep rabbit hole.

The Second Approach — Via Induced Group Actions —

My own interest in local-global symmetry is of a quite different sort.

See properties of the two patterns illustrated in a note of 24 December 1981 —

Pattern A above actually has as few  symmetries as possible
(under the actions described in the diamond theorem ), but it
does  enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part  induces
a global group action on the whole .

* For some historical background on the term "homogeneous,"
    see the Wikipedia article Homogeneous space.

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