In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

A class of closely related objects is the Dynkin diagrams, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.

A Coxeter group is a group that admits a presentation: $${\displaystyle \langle r_{0},r_{1},\dots ,r_{n}\mid (r_{i}r_{j})^{m_{i,j}}=1\rangle }$$ where the m_i,j are integers that are elements of some symmetric matrix M which has 1s on its diagonal. (Thus each generator ${\displaystyle r_{i}}$ has order 2.) This matrix M, the Coxeter matrix, completely determines the Coxeter group.

Since the Coxeter matrix is symmetric, it can be viewed as the adjacency matrix of an edge-labeled graph that has vertices corresponding to the generators r_i, and edges labeled with m_i,j between the vertices corresponding to r_i and r_j. In order to simplify these diagrams, two changes can be made:

The resulting graph is a Coxeter-Dynkin diagram that describes the considered Coxeter group.

Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), A, with matrix elements a_i,j = a_j,i = −2 cos(π/p_i,j) where p_i,j is the branch order between mirrors i and j; that is, π/p_i,j is the dihedral angle between mirrors i and j. As a matrix of cosines, A is also called a Gramian matrix. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. A is closely related to the Cartan matrix, used in the similar but directed graph: the Dynkin diagram, in the limited cases of p = 2,3,4, and 6, which are generally not symmetric.

The determinant of the Schläfli matrix is called the Schläflian;^{[citation needed]} the Schläflian and its sign determine whether the group is finite (positive), affine (zero), or indefinite (negative). This rule is called Schläfli's Criterion.^{[failed verification]}

The eigenvalues of the Schläfli matrix determine whether a Coxeter group is of finite type (all positive), affine type (all non-negative, at least one is zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use the following definitions: