Abstract Polytope - Exchange Maps

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 φ_i. These maps are called exchange maps, since they swap pairs of flags : (Ψφ_i)φ_i = Ψ always. Some other properties of the exchange maps :

• φ_i2 is the identity map
• The φ_i 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, φ_iφ_j = φ_jφ_i
• If α is an automorphism of the polytope, then αφ_i = φ_iα
• If the polytope is regular, the group generated by the φ_i 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.