Examples for Coxeter Groups

Type $I_{m}$

Type $A_{n - 1}$ ( $n \geq 2$ )

Consider $S_{n}$ act as a permutation on $R^{n}$ by permuting the standard basis vectors $e_{1}, \dots, e_{n}$ . Then the transpositions acts as reflection, sending $e_{i} - e_{j}$ to $- (e_{i} - e_{j})$ . The group formed by these flections is of type $A_{n - 1}$ Coxeter Group.

Type $B_{n}$ ( $n \geq 2$ )

Consider $S_{n}$ act as a permutation on $R^{n}$ by permuting the standard basis vectors $e_{1}, \dots, e_{n}$ . Again the transpositions act as reflection, sending $e_{i} - e_{j}$ to $- (e_{i} - e_{j})$ . Add other reflections as $e_{i} \to - e_{i}$ . The group generated by these reflections is isomorphic to $S_{n} ⋉ (Z /2 Z)^{n}$ .

Type $D_{n}$ ( $n \geq 4$ )

It’s a subgroup of $B_{n}$ with index 2 as described: Still consider $S_{n}$ as premutation on $R^{n}$ , reflections sending $e_{i} - e_{j}$ to $- (e_{i} - e_{j})$ , together with even number of signs, generated by reflections $e_{i} + e_{j} \to - (e_{i} + e_{j}), i \neq = j$ . The group generated by these reflections is isomorphic to $S_{n} ⋉ (Z /2 Z)^{n - 1}$ .

Quartz 4

Explorer

Examples for Coxeter Groups

Type $I_{m}$

Type $A_{n - 1}$ ( $n \geq 2$ )

Type $B_{n}$ ( $n \geq 2$ )

Type $D_{n}$ ( $n \geq 4$ )

Graph View

Table of Contents

Quartz 4

Explorer

Examples for Coxeter Groups

Type Im​

Type An−1​ (n≥2)

Type Bn​ (n≥2)

Type Dn​ (n≥4)

Graph View

Table of Contents

Type $I_{m}$

Type $A_{n - 1}$ ( $n \geq 2$ )

Type $B_{n}$ ( $n \geq 2$ )

Type $D_{n}$ ( $n \geq 4$ )