Exchange Maps
Let Ψ be a flag of an abstract n-polytope, and let −1 < i < n. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from Ψ by a rank i element, and the same otherwise. If we call this flag Ψ(i), then this defines a collection of maps on the polytopes flags, say φi. These maps are called exchange maps, since they swap pairs of flags : (Ψφi)φi = Ψ always. Some other properties of the exchange maps :
- φi2 is the identity map
- The φi generate a group. (The action of this group on the flags of the polytope is an example of what is called the flag action of the group on the polytope)
- If |i − j| > 1, φiφj = φjφi
- If α is an automorphism of the polytope, then αφi = φiα
- If the polytope is regular, the group generated by the φi is isomorphic to the automorphism group, otherwise, it is strictly larger.
The exchange maps and the flag action in particular can be used to prove that any abstract polytope is a quotient of some regular polytope.
Read more about this topic: Abstract Polytope
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