Monodromy group

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The Cayley graph of the monodromy group of a tetrahedron

The monodromy group of an abstract polytope, 𝓟, written Mon(𝓟), the group generated by flag changes.

Definition[edit | edit source]

For a polytope 𝓟 of rank n , let σ i  be the bijection which maps each flag of 𝓟 to its i -adjacent flag. Then the monodromy group is σ 0 ,σ 1 ,...,σ n -1 , that is the group generated by flag changes.

Properties[edit | edit source]

  • A polytope is regular iff its monodromy group is isomorphic to its automorphism group. Furthermore there is an isomorphism between the two which maps the flag changes to the distinguished generators.
  • The monodromy group of a polyhedron is the automorphism group of its minimal regular cover.[1] In ranks greater than 3 this is not always the case - the tomotope is an example.

References[edit | edit source]

Bibliography[edit | edit source]

  • Monson, Barry; Pellicer, Daniel; Williams, Gordan (2012), "The Tomotope", Ars Mathematica Contemporanea