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3.4 Affine Coxeter groups

Affine Coxeter groups are by definition such that the bilinear form B is positive semi-definite but not positive definite. The radical V ⊥ (defined as the subspace of vectors x for which B (x,y ) ≡ xT By = 0 for all y) is then one-dimensional (in the irreducible case). Indeed, since B is positive semi-definite, its radical coincides with the set N of vectors such that λT B λ = 0 as can easily be seen by going to a basis in which B is diagonal (the eigenvalues of B are non-negative). Furthermore, N is at least one-dimensional since B is not positive definite (one of the eigenvalues is zero). Let μ be a vector in V ⊥ ≡ N. Let ν be the vector whose components are the absolute values of those of μ, νi = |μi|. Because Bij ≤ 0 for i ⁄= j (see definition of B in Equation (3.28View Equation)), one has
0 ≤ νT B ν ≤ μT B μ = 0

and thus the vector ν belongs also to ⊥ V. All the components of ν are strictly positive, νi > 0. Indeed, let J be the set of indices for which νj > 0 and I the set of indices for which νi = 0. From ∑ j Bkj νj = 0 (ν ∈ V⊥) one gets, by taking k in I, that Bij = 0 for all i ∈ I, j ∈ J, contrary to the assumption that the Coxeter system is irreducible (B is indecomposable). Hence, none of the components of any zero eigenvector μ can be zero. If ⊥ V were more than one-dimensional, one could easily construct a zero eigenvector of B with at least one component equal to zero. Hence, the eigenspace ⊥ V of zero eigenvectors is one-dimensional.

Affine Coxeter groups can be identified with the groups generated by affine reflections in Euclidean space (i.e., reflections through hyperplanes that may not contain the origin, so that the group contains translations) and have also been completely classified [107Jump To The Next Citation Point]. The translation subgroup of an affine Coxeter group ℭ is an invariant subgroup and the quotient ℭ0 is finite; the affine Coxeter group ℭ is equal to the semi-direct product of its translation subgroup by ℭ0. We list all the affine Coxeter groups in Table 2.


Table 2: Affine Coxeter groups.

Name

Coxeter graph

A+1

PIC

   

A+n (n > 1)

PIC

   

B+n (n > 2 )

PIC

   

+ Cn

PIC

   

+ Dn

PIC

   

G+ 2

PIC

   

+ F4

PIC

   

E+6

PIC

   

E+7

PIC

   

E+8

PIC



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