Capturing Polytopal Symmetries by Coloring the Edge-Graph

Throughout the text we let $P\subset\mathbb{R}^{d}$ denote a convex polytope that is full-dimensional (i.e., not contained in any proper affine subspace of $\mathbb{R}^{d}$) and contains the origin in its interior (i.e., $0\in\operatorname{int}(P)$).

By $\mathcal{F}_{\delta}(P)$ we denote the set of $\delta$-dimensional faces of $P$. We assume a fixed enumeration $v_{1},...,v_{n}\in\mathcal{F}_{0}(P)$ of the polytope’s vertices. In particular, $n$ will always denote the number of the vertices.

The edge-graph of $P$ is the finite simple graph $G_{P}=(V,E)$ with vertex set $V=\{1,...,n\}$ and edge set $\smash{E\subseteq{V\choose 2}}$. We implicitly assume that $i\in V$ corresponds to the vertex $v_{i}\in\mathcal{F}_{0}(P)$, and that $ij\in E$ (short for $\{i,j\}\in E$) if and only if $\operatorname{conv}\{v_{i},v_{j}\}\in\mathcal{F}_{1}(P)$.

The (combinatorial) symmetry group of $G_{P}$¹¹1For convenience, notions like the symmetry group, colorings, the adjacency matrix, etc.  are only introduced for the edge-graph, but it is understood that they apply to more general graphs as well. is defined as

that is, the group of permutations of $V$ that fix the edge set of $G_{P}$.

A coloring of $G_{P}$ is a map $\mathfrak{c}\colon V\mathbin{\mathaccent 0{\cdot}\cup}E\to\mathfrak{C}$ (it assign colors to both, vertices and edges), where $\mathfrak{C}$ denotes an abstract set of colors. The pair $(G_{P},\mathfrak{c})$ is then a colored edge-graph and will be abbreviated by $G_{P}^{\mathfrak{c}}$. Its combinatorial symmetry group is

If $\sigma\in\operatorname{Aut}(G_{P}^{\mathfrak{c}})$, we also say that $\sigma$ preserves the coloring $\mathfrak{c}$.

The colored adjacency matrix of $G_{P}^{\mathfrak{c}}$ is the matrix $A^{\mathfrak{c}}\in(\mathfrak{C}\mathbin{\mathaccent 0{\cdot}\cup}\{0\})^{n\times n}$ with entries

Clearly, a coloring is completely determined by the colored adjacency matrix, and we might occasionally use $A^{\mathfrak{c}}$ to define a coloring.

A geometric symmetry $T\in\operatorname{Aut}_{\operatorname{GL}}(P)$ of $P$ maps vertices of $P$ onto vertices of $P$ and thus describes a permutation of the vertex set. Let $\sigma_{T}\in\operatorname{Sym}(V)$ be the permutation of the vertex set of the edge-graph that permutes its vertices in the same way as $T$ permutes the vertices of $P$. Formally, that is

Since $T$ also maps edges of $P$ onto edges of $P$, also $\sigma_{T}$ maps edges to edges, and so we see that $\sigma_{T}$ is a symmetry of the edge-graph, i.e., $\sigma_{T}\in\operatorname{Aut}(G_{P}$). The assignment $T\mapsto\sigma_{T}$ then defines a group homomorphism $\phi\colon\operatorname{Aut}_{\operatorname{GL}}(P)\to\operatorname{Aut}(G_{P})$ which we shall call the natural group homomorphism of the polytope $P$.

Since $P$ is full-dimensional, its vertices contain a basis of $\mathbb{R}^{d}$, and it follows that $\phi$ must be injective. In general however, $\phi$ is not an isomorphism and $\operatorname{Aut}_{\operatorname{GL}}(P)\not\cong\operatorname{Aut}(G_{P})$, which is a formal way to say that the edge-graph $G_{P}$ can have many more symmetries than the polytope.

Our approach for rectifying this is to assign a coloring $\mathfrak{c}\colon V\mathbin{\mathaccent 0{\cdot}\cup}E\to\mathfrak{C}$ to the edge-graph $G_{P}$ with the hope that $\operatorname{Aut}_{\operatorname{GL}}(P)\cong\operatorname{Aut}(G_{P}^{\mathfrak{c}})$. The natural candidate for the isomorphism between the groups is a colored version of the natural homomorphism:

For this to work as desired, we need to check two things:

• •

  First, $\phi^{\mathfrak{c}}$ needs to be well-defined. This is not the case for each coloring: one needs to check that for each $T$ $\in$ $\operatorname{Aut}_{\operatorname{GL}}(P)$ the corresponding permutation $\sigma_{T}$ is indeed a symmetry of $G_{P}^{\mathfrak{c}}$ (that is, is in $\operatorname{Aut}(G_{P}^{\mathfrak{c}})$). Intuitively, this amounts to checking that the edge-graph, even after coloring, is still at least as symmetric as $P$.

• •

  Second, $\phi^{\mathfrak{c}}$ must have an inverse. If so, then $G_{P}^{\mathfrak{c}}$ is exactly as symmetric as $P$. Providing such an inverse will go as follows: for each $\sigma\in\operatorname{Aut}(G_{P}^{\mathfrak{c}})$ we need to construct a geometric symmetry $T_{\sigma}\in\operatorname{Aut}_{\operatorname{GL}}(P)$ with

  $$T_{\sigma}v_{i}=v_{\sigma(i)},\quad\text{for all $i\in V$}.$$

  Since $P$ is full-dimensional, if $T_{\sigma}$ exists then it is unique. The map $\sigma\mapsto T_{\sigma}$ is then the desired inverse.

The discussion also applies verbatim to the orthogonal symmetry group $\operatorname{Aut}_{\operatorname{O}}(P)$, and we shall use the same notation $\phi^{\mathfrak{c}}\colon\operatorname{Aut}_{\operatorname{O}}(P)\to\operatorname{Aut}(G_{P}^{\mathfrak{c}})$ to denote the natural homomorphism in this case.

With this in place, we can formalize “capturing symmetries”:

The main results of this article are explicit constructions for colorings that

• •

  capture linear symmetries (Theorem˜4.7).

• •

  capture orthogonal symmetries (Theorem˜5.2).

In Section˜2 we introduce the metric coloring and the orbit coloring, two very natural candidates for capturing certain polytopal symmetries. In this section we do not yet show that either coloring capture linear or orthogonal symmetries, but we establish relevant properties used in the upcoming sections.

In Section˜3 we derive a sufficient condition for a coloring of the form $\mathfrak{c}\colon V\mathbin{\mathaccent 0{\cdot}\cup}E\to\mathbb{R}$ (the colors are real numbers) to capture linear symmetries. The criterion will be in terms of the eigenspaces of the (colored) adjacency matrix of the edge-graph. We shall call this the “linear algebra criterion”.

In Section˜4 we introduce the Izmestiev coloring (based on a construction by Ivan Izmestiev [4]) and we show that it satisfies the “linear algebra criterion” from Section˜3. We thereby establish the existence of a first coloring that captures linear symmetries (Theorem˜4.7). As a corollary we find that the orbit coloring captures linear symmetries as well (Corollary˜4.8).

In Section˜5 we show that a combination of the Izmestiev coloring and the metric coloring captures orthogonal symmetries (Theorem˜5.2).

Our first coloring is motivated from the previously mentioned construction of Bremner et al. [3] – a coloring of the complete graph $K_{n}$ that “captures orthogonal symmetries”. In our notation their result reads as follows:

The strength of this result lies in its immediate applicability: constructing this “complete metric coloring” requires no knowledge of the edge-graph (which is usually hard to come by), but only the vertex coordinates of $P$ ⁴⁴4If $P$ is given in $\mathcal{H}$-representation, one can apply Theorem 2.1 to compute the orthogonal symmetry group of the dual polytope $P^{\circ}$, which is identical to $\operatorname{Aut}_{\operatorname{O}}(P)$ as a matrix group.. In practice, this is probably the best tool for an explicit computation of $\operatorname{Aut}_{\operatorname{O}}(P)$.

From a theoretical and aesthetic perspective however, this construction has the flaw of containing massively redundant data and stepping outside the combinatorial structure of the polytope (we assign color to vertex-pairs that are not edges of the polytope). Naturally, we can ask whether one can get away with coloring fewer of these “non-edges”, ideally only the actual edges of the edge-graph.

Based on this hope, we define the following:

Whether the metric coloring captures orthogonal symmetries is an open question (see also ˜6.6). Our reason for introducing it anyway is that in Section˜5 the metric coloring will be one ingredient to a coloring that indeed captures orthogonal symmetries.

We close this section with another formulation of Theorem˜2.1 that also allows for capturing linear symmetries (in fact, this is closer to the original formulation in [3]). Note that the complete metric coloring of $K_{n}$ in Theorem˜2.1 can also be described by its colored adjacency matrix $A^{\mathfrak{c}}=\Phi^{\top}\Phi$, where $\Phi:=(v_{1},...,v_{n})\in\mathbb{R}^{d\times n}$ is the matrix in which the vertex coordinates of $P$ appear as columns.

A proof for part (ii) will also follow from the theory developed in Section˜3 (see Remark˜3.2)

The next coloring is motivated from the following consideration: suppose that we are given two vertices $v_{i},v_{j}\in\mathcal{F}_{0}(P)$ in the same orbit w.r.t. $\operatorname{Aut}_{\operatorname{GL}}(P)$, which just means that there is a $T\in\operatorname{Aut}_{\operatorname{GL}}(P)$ with $Tv_{i}=v_{j}$. The corresponding combinatorial symmetry $\sigma_{T}\in\operatorname{Aut}(G_{P})$ satisfies $\sigma_{T}(i)=j$. If now $\mathfrak{c}\colon V\mathbin{\mathaccent 0{\cdot}\cup}E\to\mathfrak{C}$ is a coloring that captures linear symmetries, then $\sigma_{T}$ preserves the coloring $\mathfrak{c}$ and we have $\mathfrak{c}(j)=\mathfrak{c}(\sigma_{T}(i))=\mathfrak{c}(i)$. We can summarize this as follows: if $\mathfrak{c}$ is supposed to capture linear symmetries, then vertices in the same $\operatorname{Aut}_{\operatorname{GL}}(P)$-orbit of $P$ must have the same color in $G_{P}^{\mathfrak{c}}$. With an analogous argument we see that the same holds for edges.

Having identified this first necessary condition for capturing symmetries, we can consider the “simplest” coloring that follows this idea:

An analogous coloring can be defined for orthogonal symmetries, which we shall call the orthogonal orbit coloring of $G_{P}$, still denoted by $\mathfrak{o}$. For the sake of conciseness, this section only discusses the (linear) orbit coloring, but all statements carry over to the orthogonal version in the obvious way.

As we shall learn in Section˜4 (see Corollary˜4.8), the orbit coloring indeed captures linear symmetries. However, this is surprisingly hard to show directly. In fact, our eventual proof of this will “just” use the following:

Besides linear and orthogonal symmetries, there are at least two further common groups of symmetries associated with a polytope: the projective symmetries and the combinatorial symmetries (that is, the symmetries of the face lattice).

We can ask whether those too can be captured by a colored edge-graph:

There might be a general strategy derived from the following (informal) inclusion chain of the symmetry groups:

As it turns out, having solved the coloring problem further to the left in the chain can help to solve the problem further to the right – at least to some degree.

For example, note that every polytope $P$ can be linearly transformed via a transformation $T\in\operatorname{GL}(\mathbb{R}^{d})$ so that $\operatorname{Aut}_{\operatorname{GL}}(P)=\operatorname{Aut}_{\operatorname{O}}(TP)$. That is, a coloring of $G_{P}$ that captures the orthogonal symmetries of $TP$ (which has the same edge-graph) also captures the linear symmetries of $P$. In still other words, we solved the problem of capturing linear symmetries by making use of our ability to capture orthogonal symmetries.

In our approach, we have not made use of this because we needed to solved the linear case before the orthogonal one. However, this can be of use for capturing projective symmetries. More explicitly, the question is as follows: for every polytope $P$, is there a projective transformation $T\in\operatorname{PGL}(\mathbb{R}^{d})$ so that $\operatorname{Aut}_{\operatorname{PGL}}(P)=\operatorname{Aut}_{\operatorname{GL}}(TP)$?

The same approach seems doomed for capturing combinatorial symmetries: there are polytopes with combinatorial symmetries that cannot be realized geometrically (​[1] discusses the case of a combinatorial symmetry that cannot be made linear; to our knowledge, realizing them as projective symmetries remains to be discussed).

For capturing the symmetries of certain 2-dimensional polytopes it is necessary to color both vertices and edges (cf. Figure˜2). But it is unclear whether this is still necessary in higher dimensions.

A vertex-only coloring is not always sufficient. For example, in even dimensions exist vertex-transitive neighborly polytopes other than the simplex: e.g. for $n\geq 6$ we have the following cyclic 4-polytope with $n$ vertices that is not a simplex:

The edge-graph of $P$ is the complete graph $K_{n}$, and $P$ has a single orbit of vertices. Thus, if $\mathfrak{c}\colon V\to\mathfrak{C}$ is a vertex-only coloring that captures the symmetries of $P$, then all vertices of $K_{n}$ must receive the same color. But if the edges receive no color, then $\operatorname{Aut}(K_{n}^{\mathfrak{c}})$ $=\operatorname{Sym}(V)$. However, it is known that the linear symmetry group of the cyclic polytope $P$ other than a simplex is strictly smaller than $\operatorname{Sym}(V)$ [5].

Our approach suggests no immediate generalization to non-convex polytopes or various forms of polytopal complexes.

A vast generalization of polytope skeleta are graph embeddings. For a graph $G=$ $(V,E)$, a graph embedding is simply a map $v\colon V\to\mathbb{R}^{d}$. There are natural notions of symmetry for such embeddings, and so one might ask whether it is possible to “capture” them by coloring the graph. The following example shows that this is not possible in general: