Complex polytope

Not to be confused with Polytope complex.

A complex polytope is a generalized polytope-like object whose containing space is n -dimensional complex coordinate space $\mathbb {C} ^{n}$. Complex coordinate space is an extension of Euclidean space $\mathbb {R} ^{n}$ where each dimension has a real and imaginary axis. Complex polytopes are in general not actually polytopes in the traditional sense, since they violate dyadicity, but are rather a kind of incidence complex together with a realization. Another major difference from real polytopes is that complex polytopes don't enclose points and don't have interiors.

Complex space

See also: w:Complex coordinate space

Complex 1-space, that is the vector space formed by the complex numbers themselves $\mathbb {C} ^{1}$, is sometimes called the complex line and other times the complex plane. The former name reflects that it is of dimension 1 over $\mathbb {C}$, while the latter reflects that it's of dimension 2 over $\mathbb {R}$.

To define symmetries of complex space we first define a bilinear form called the Hermetian form:

$$\langle (x_{0},x_{1},\dots ,x_{n})\mid (y_{0},y_{1},\dots ,y_{n})\rangle =\sum _{i=0}^{n}x_{i}{\overline {y_{i}}}$$

Here b i  represents the component of b  over some arbitrary choice of orthonormal basis, and b represents the complex conjugate.

The vector space ℂn  equipped with the Hermetian form gives an inner product space called unitary space. A unitary matrix is a matrix that preserves the Hermetian form.

Each isometry of unitary space is composed of a unitary matrix and a translation. For finite complex polytopes their symmetries will only ever be linear transformations and thus they can be represented by a unitary matrix. Isometries with a non-trivial translational component only appear as symmetries of apeirotopes.

By this definition, complex n -space as a metric space is isometric to 2n -dimensional Euclidean space. In fact there is a simple isomorphism between them which preserves the metric. However, the set of linear transformations is more restricted in complex n -space than in 2n -dimensional Euclidean space. Transformations such as complex conjugation are nonlinear in complex space but linear when mapped to Euclidean space.

Definition

Incidence

Let 𝓟 be a set of affine subspaces of the unitary space $\mathbb {C} ^{n}$. We introduce some terminology:

• μ -dimensional elements of 𝓟 are called μ -flats.
• Two flats are incident if one is a proper subspace of the other.
• For any two flats F  and H , the medial figure of F  and H  is the set of all flats G  such that $F\subsetneq G\subsetneq H$.
• A set of flats 𝓧 is connected if for any two flats, F  and H , in 𝓧 there is a sequence of flats, S , beginning with F  and ending with H , such that every pair of consecutive flats in S  are incident.

Then 𝓟 is a complex polytope iff it satisfies the following properties:

• There is a $-1$-flat consisting of the empty space, and an n -flat consisting of $\mathbb {C} ^{n}$.
• Every medial figure contains at least two elements.
• If $\lambda <\nu -2$ then the medial figure of a λ -flat and a ν -flat is connected.

In other words the abstract structure of P  is an incidence complex.

Distinguished generators

Regular complex polytopes can be defined in terms of an indexed list of group generators, much the way abstract regular polytopes and regular skew polytopes have distinguished generators. The distinguished generators of a complex polytope however have relaxed requirements to reflect the non-dyadicity of complex polytopes.

In an n -dimensional unitary space V , a pseudo-reflection is a linear map $\rho :V\rightarrow V$, such that:

• ρ  has finite order
• The fixed points of ρ  form a n-1 -dimensional linear subspace. This space is called the reflecting hyperplane of ρ 

A rank-n  regular complex polytope is a sequence of pseudo-reflections in $\mathbb {C} ^{n}$, $\langle \rho _{0},\rho _{1},\rho _{2}\rangle$, such that:

• If $|i-j|>1$ then $\rho _{i}\rho _{j}=\rho _{j}\rho _{i}$
• For each $0\leq i<n-1$, there is a positive integer $q_{i}$ such that $\underbrace {\rho _{i}\rho _{i+1}\rho _{i}\dots } _{q_{i}}=\underbrace {\rho _{i+1}\rho _{i}\rho _{i+1}\dots } _{q_{i}}$

As with other distinguished generator based constructions, elements of the group generated by the distinguished generators correspond to flags. The concrete vertex locations are determined by choosing an arbitrary seed vertex on the intersection of the reflecting hyperplanes of $\rho _{i}$ other than $\rho _{0}$, and applying every group element to the seed to obtain the remainder.

Complex 1-polytopes

Complex 1-polytopes (polytela) may have two vertices or more, so dyadicity is already violated. Their vertices are simply located at distinct points in $\mathbb {C} ^{1}$.

Real 1-polytopes enclose an interval of points as a consequence of the ordering of the real numbers. But as the complex numbers don't have an analogue of this ordering, it isn't meaningful to speak of the interior of a complex 1-polytope, nor a complex polytope of any higher rank.

There is exactly one regular complex 1-polytope with n  vertices for each n  ≥ 2, and its vertices are located at the n th roots of unity (or any combination of isometries and uniform scalings thereof), forming the vertices of a regular polygon in its Argand diagram.

Complex polygons

Complex polygons have vertices located in $\mathbb {C} ^{2}$. While real edges connect two points in a line, a complex edge of degree n  (a.k.a. an n -edge) connects n  ≥ 2 points that live in an affine complex subspace of dimension 1. This is the complex space equivalent of the planarity condition for Euclidean polytopes. Each point must be adjacent to 2 or more edges, and the number of edges is also called its degree.

The regular complex polygons were completely characterized in 1991 by Coxeter. All the edges must have the same degree and the same goes for the vertices, so regular complex polygons are abstractly a type of configuration. Clearly, all edges of a regular complex polygon are themselves regular and congruent to each other.

External links

• Wikipedia contributors. "Complex polytope".

Bibliography

• Coxeter, Donald (1974). Regular complex polytopes. ISBN 0521394902.
• Orlik, Peter; Reiner, Victor; Shepler, Anne (2002). "The sign representation for Shephard groups" (PDF). Mathematische Annalen. 322 (3): 477–492. arXiv:math/0011105. doi:10.1007/s002080200001.