Schläfli matrix

A Schläfli matrix, sometimes called a Gram matrix, is a particular matrix associated to a given reflection group. Suppose that a reflection group is generated by reflections by mirrors with normals $$m_1,\ldots,m_n$$. Let $$a_{ij}$$ be the cosine of the angle between the normals $$m_i$$ and $$m_j$$. This is the supplement of the angle between the mirrors. Then, the associated Schläfli matrix is given by


 * $$\begin{bmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\ldots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{1n}&a_{2n}&\ldots&a_{nn}\end{bmatrix}.$$

The Schläfli matrix will be symmetric and have ones along its main diagonal. Note that some authors define the Schläfli matrix as twice this matrix, which greatly simplifies hand calculations.

One may also think of a Schläfli matrix as describing the angles of a simplex in either spherical, Euclidean, or hyperbolic space. From this point of view, the Schläfli matrix of a reflection group is that of its fundamental domain.

Schläfli matrices are helpful to compute properties of Wythoffian polytopes. Schläfli's criterion gives a necessary condition on this matrix for the reflection group to be embedabble on Euclidean space. The inverse of the Schläfli matrix, which has been called the Stott matrix by Wendy Krieger, may be used to compute the circumradius of a Wythoffian polytope from its fractional Coxeter diagram.

Schläfli's criterion
Schläfli's criterion states that the determinant of this matrix must be non-negative whenever the reflection group is embedded in Euclidean space, and zero when it describes a group with translations. The original form of this criterion was stated by Coxeter, though it applied only to reflection groups with linear diagrams. This slightly more general form may be proved as follows. One constructs the matrix


 * $$M=\begin{bmatrix}\vec m_1,\vec m_2,\ldots,\vec m_n\end{bmatrix}$$

from the unit normal vectors of the mirrors of the reflection group. From basic properties of the dot product, the Schläfli matrix is given precisely by $$M^TM$$, whose determinant $$\det(M)^2$$ will be non-negative, and non-zero as long as the normals are linearly independent.

Note that the converse of this criterion does not hold. For instance, the Schläfli matrix for the Coxeter group o7o3o o7o3o has positive determinant, despite not being possible to embed in Euclidean space (since o7o3o can't be embedded either).

Circumradius formula
Wendy Krieger gives the following method to calculate the circumradius of a polytope from its fractional Coxeter diagram. One takes the node vector $$\vec r$$ of distances associated to the nodes of the diagram, containing a 1/2 for ringed nodes, 0 for unringed nodes, etc. The circumradius $$R$$ is then given by


 * $$R^2=\vec{r}^T S^{-1}\vec{r}=\sum_{i=1}^n\sum_{j=1}^n S^{-1}_{ij}r_ir_j,$$

where $$S$$ is the Schläfli matrix of the diagram, and $$\vec{r}^T$$ is the transpose.

The reason this formula works is as follows. Consider the ordered basis $$\mathcal M=(\vec m_1,\vec m_2,\ldots,\vec m_n)$$ of unit mirror normals, and the dual base $$\mathcal B=(\vec b_1,\vec b_2,\ldots,\vec b_n)$$ such that $$\vec m_i\cdot\vec b_j$$ equals 1 if $$i=j$$ and 0 otherwise. The node vector is written in terms of the basis $$\mathcal B$$. Being a bilinear form, the dot product is determined from its values on the basis $$\mathcal M$$ as


 * $$\vec v_1\cdot\vec v_2=\vec v_1^T S \vec v_2,$$

where $$S$$ is the matrix whose $$(i,j)$$ entry is given by $$\vec m_i\cdot\vec m_j$$. This is precisely the Schläfli matrix. Due to elementary properties of the dot product, it may be seen that the change of basis matrix between $$\mathcal M$$ and $$\mathcal B$$ is also the Schläfli matrix. As a result, if $$\vec r$$ is a generator for the polytope, written in terms of the basis $$\mathcal B$$, its magnitude may be computed as


 * $$\|\vec r\|^2=\vec r\cdot\vec r=(S^{-1}\vec r)^T S(S^{-1}\vec r)=\vec{r}^T S^{-1} \vec{r}.$$

Hyperbolic space
As not all hyperplanes intersect in hyperbolic space, one must take some care to define the Schläfli matrix. It turns out the best way to do this is to define the entry $$a_{ij}$$ corresponding to two non-intersecting hyperplanes as the hyperbolic cosine of their distance. Due to the identity $$\cosh(x)=\cos(ix)$$, one might say equivalently say that these mirrors make an imaginary angle. One may notate this mirror arrangement via a fractional Coxeter diagram using pure imaginary numbers.

Using this convention, the circumradius formula works the same for hyperbolic polytopes. Recall the hyperboloid model of hyperbolic space, where one embeds hyperbolic space $$\mathbb H^n$$ in a vector space $$\mathbb R^{n+1}$$ with a certain quadratic form $$Q$$. Hyperbolic space is one of the connected components of the locus of $$Q(x)=-r$$ for some $$r>0$$. In analogy to the spherical case, we may define its circumradius as $$\sqrt{-r}=\sqrt{r}i$$. By replacing the dot product by the bilinear form $$\circ$$ defined from $$Q$$ in the previous argument, one may prove that the analogous formula in hyperbolic space works just as well, via the identity


 * $$a\circ b=|a||b|\cos(\theta)$$

for mirrors at a possibly imaginary angle $$\theta$$.