# Quotient polytope

(Redirected from Quotient)

## Definition[edit | edit source]

Let 𝓟 be an abstract polytope, with a partial order ≤_{𝓟}. Let Γ (𝓟) be the automorphism group acting on the elements of 𝓟. Let Σ be a subgroup of Γ (𝓟) with the restricted action on the elements of 𝓟.

We denote the set of orbits on the elements of 𝓟 as 𝓟/Σ . We call the function which maps an element of 𝓟 to it's orbit under Σ the **canonical projection map**, and denote it π .

We introduce a partial order, ≤_{𝓟/Σ }, on 𝓟/Σ such that F ≤_{𝓟/Σ } G iff there exists some f in π -1 [F] and g in π -1 [G], such that f ≤_{𝓟} g .

The set 𝓟/Σ along with the partial order ≤_{𝓟/Σ } is called the **quotient** of 𝓟 with respect to Σ . If it is also a polytope it is called a **quotient polytope**.

## Regular quotients[edit | edit source]

This section is empty. You can help by adding to it. |

## Bibliography[edit | edit source]

- McMullen, Peter; Schulte, Egon (December 2002).
*Abstract Regular Polytopes*. Cambridge University Press. ISBN 0-521-81496-0.

This article is a stub. You can help Polytope Wiki by expanding it. |