Coxeter and Dynkin Diagrams: Structure & Applications

Updated 7 January 2026

Coxeter and Dynkin diagrams are graphical tools for encoding generating reflections and the structure of root systems in finite reflection groups.
They facilitate classification by revealing subgroup hierarchies, parabolic structures, and symmetries critical in invariant theory and polytope geometry.
These diagrams underpin computational group theory and operator methods, linking diagram automorphisms to extended group symmetries and cohomological applications.

Coxeter and Dynkin diagrams encode the combinatorics of generating reflections and the multiplication rules in finite reflection groups, both real and complex. These diagrams provide a graphical mechanism for presenting the relations in Coxeter groups and describing the structure of the underlying root systems. Their applications permeate classification theory, invariant theory, geometry of polytopes, operator theory, and representation theory. This exposition synthesizes technical foundations, classification principles, algebraic interpretations, geometric links, and computational and cohomological applications, as established in foundational works (Douglass et al., 2011, Taylor, 2012, Somberg, 2013, Lange et al., 2015, Gandhi, 2019, Hohlweg, 2011).

1. Definitions and Algebraic Presentation

A finite reflection group $W$ is a subgroup of $\mathrm{GL}(V)$ (for $V$ a real or complex vector space) generated by (pseudo-)reflections: elements $r\in \mathrm{GL}(V)$ fixing a hyperplane pointwise with finite order (Somberg, 2013, Taylor, 2012). If $V$ is equipped with a nondegenerate inner product, reflections act orthogonally, defined by $s_\alpha(v) = v - 2\frac{(v,\alpha)}{(\alpha,\alpha)}\alpha$ , where $\alpha$ is a root and $s_\alpha$ fixes $\alpha^{\perp}$ (Lange et al., 2015). The collection of simple roots $\Delta = \{\alpha_1,\dots, \alpha_n\}$ determines generators $s_i = s_{\alpha_i}$ and the group presentation

$W = \langle s_1, \dots, s_n \mid (s_i s_j)^{m_{ij}} = 1 \rangle,$

with $m_{ii}=1$ , $m_{ij}=m_{ji}\geq 2$ for $i\neq j$ . The integers $m_{ij}$ are encoded as edges in the Coxeter diagram: vertices correspond to $s_i$ ; edges between $i$ and $j$ are labeled $m_{ij}\ge3$ if $m_{ij}>2$ ; edges are omitted for $m_{ij}=2$ (Douglass et al., 2011, Gandhi, 2019).

The Dynkin diagram is a refinement for crystallographic cases, labeling edges according to angles between roots (product of Cartan integers). These diagrams uniquely determine the group up to conjugation.

2. Classification via Diagrams

Irreducible finite reflection groups are classified by their Coxeter and Dynkin diagrams (Somberg, 2013, Douglass et al., 2011, Lange et al., 2015). The crystallographic ("Weyl") types are $A_n$ , $B_n$ , $C_n$ , $D_n$ , $E_6$ , $E_7$ , $E_8$ , $F_4$ , $G_2$ ; noncrystallographic types include $H_3$ , $H_4$ , and $I_2(m)$ (dihedral). The diagram determines both the group structure and the geometry of the associated root system. Each node is a simple reflection; edges (with labels) encode the order of $s_i s_j$ . Deletion of nodes yields parabolic subgroups.

For complex reflection groups, the Shephard–Todd classification assigns diagrams to the exceptional groups $G_k$ , $4 \le k \le 37$ , with Coxeter-type diagrams for those whose subgroups can be realized by reflection and rotation (Taylor, 2012, Lange et al., 2015). The diagram automorphisms correspond to normalizing rotations in the group extensions (Lange et al., 2015).

3. Reflection Subgroups and Parabolic Structure

Every finite Coxeter group admits reflection subgroups generated by subsets of its reflections. Parabolic subgroups are defined as pointwise stabilizers of subspaces and are themselves reflection subgroups (Douglass et al., 2011, Taylor, 2012). Any subset of simple reflections generates a parabolic subgroup: for $W=\langle r_1,\dots,r_n\rangle$ , every $S\subseteq \{r_1,\dots,r_n\}$ gives $\langle S\rangle$ parabolic (Taylor, 2012). The classification of reflection subgroups proceeds by listing all conjugacy classes of parabolics, then the maximal-rank reflection subgroups of each (Douglass et al., 2011). The Borel–De Siebenthal algorithm applies in Weyl cases, recursively constructing subsystems by extended Dynkin diagram deletions (Douglass et al., 2011).

The simple-extension principle organizes all reflection subgroups as chains $H_0 < H_1 < \cdots < H_k=G$ via successive addition of reflections, uniquely characterized in the associated diagram poset (Taylor, 2012).

4. Geometric and Combinatorial Interpretation

Coxeter and Dynkin diagrams encode the geometry of the underlying polytopes: permutahedra and generalized associahedra (Hohlweg, 2011). For a finite reflection group $W$ , the permutahedron $\mathrm{Perm}^{\mathbf a}(W)$ is the convex hull of the $W$ -orbit of a generic $\mathbf a\in V$ , a simple $n$ -polytope whose facet inequalities are determined by the simple roots (Hohlweg, 2011). The diagram classifies the normal fan of the polytope and supports the construction of the Cambrian fan and the cluster complex, central in the theory of cluster algebras.

Generalized associahedra correspond to oriented diagram paths following Coxeter elements, with vertices indexed by $c$ -clusters and faces by compatible root subsets. Diagram combinatorics underlie enumeration formulas for $f$ - and $h$ -vectors, Catalan numbers, and the structure of the normal fan.

5. Coxeter Representation and Operator Theory

The diagram encodes the multiplication relations critical in operator theory associated with reflection groups. Demazure operators $\Delta_i$ act on the formal group ring $R[[M]]_F$ as

$\Delta_i(u) = \frac{u - s_i(u)}{x_{\alpha_i}},$

with Leibniz-type and twisted commutation properties (Gandhi, 2019). The formal affine Demazure algebra $\mathcal{D}_F$ is generated by the diagram-defined Demazure elements $X_i$ and the associated scalar ring. The relations reflect the diagram:

Twisted commutation $X_i q = s_i(q) X_i + \Delta_i(q)$
Quadratic relations involving $X_i^2$
Braid relations as dictated by edge labels $m_{ij}$ :

$\underbrace{X_iX_j \cdots}_{m_{ij}} - \underbrace{X_jX_i \cdots}_{m_{ij}} = \sum_{w < w_0(i,j)} c_{i,j}(w) X_w$

with structure coefficients determined by diagram-induced orbits.

These operator relations interpolate between classical nil-Hecke and 0-Hecke relations under specialization of the formal group law, capturing diagram geometry in cohomological invariants (Gandhi, 2019).

6. Advanced Applications and Computational Aspects

The complete classification of reflection subgroups and their inclusions via diagram structure has profound applications in invariant theory (fixed-point sets, discriminant invariants), geometry of orbifolds and manifolds (quotients stratified by diagram), and conformally invariant differential–difference operator theory (Dunkl–Laplace deformation of GJMS operators) (Somberg, 2013).

In computational group theory, listing conjugacy classes proceeds via deletion algorithms on diagrams, as realized in GAP+CHEVIE (Douglass et al., 2011). The associated tables enumerate reflection subgroups, their diagrams, and Coxeter elements, enabling checks of properties such as injectivity and surjectivity of the map from reflection subgroups to element conjugacy classes.

7. Diagram Automorphisms, Extension Groups, and Orbifold Geometry

Diagram automorphisms induce extension mechanisms: normalizing rotations lifting to group automorphisms correspond to symmetries of the Coxeter and Dynkin diagrams (Lange et al., 2015). Extensions $G=\langle W, h\rangle$ involve an extra generator $h$ of order matching the automorphism, with relations $h s_i h^{-1}=s_{\sigma(i)}$ , capturing the automorphism action in the presentation.

For reflection–rotation groups, diagram stratification governs orbifold structures in the geometric quotient $\mathbb R^n/G$ . Local charts are cones on spherical Coxeter orbifolds or manifolds, with singularities and isotropy stratification informed by diagram (Lange et al., 2015).