Schläfli matrix

A Schläfli 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 $$m_j$$. Then, the associated Schläfli matrix is given by


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

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 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 o4o4o4*a has positive determinant, despite not being possible to embed in Euclidean space.