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The relation above ensures that s has evaluation 1 on the isotropic vector (0,1). 7 that we can find a point es E E with ev(f,es) = s(f) for all f E F. ,en define the simplex we are looking for. 11 Two simplices D and P in Euclidean space E are similar if and only if they have the same Coxeter matrix. Proof We have already seen that two similar simplices have the same Coxeter matrices. ,n. It remains to prove that p(Q) > 0. 9 to get that )0m0 + Alm1 +... Dev(m0,p) +... + µ(o,) which shows that u(o) > 0 as required.

14 we have the following explicit formulas trz 1 1 0 1 ]=4 a 0 0 1 tr2 , aEC-{0,1} 1=2+a+a-1 Notice, that the value of a+ a-' is unchanged if a is replaced by a-1. On the other hand we have the matrix formulas 0 -1 a 0 0 1 1 0 0 1 -1 0 The remaining details are left to the reader. 1 Let E be a finite dimensional vector space equipped with a non- singular quadratic form. 10 Let F be a non-singular subspace of E, show that E = F ® F 1 and that (x,y)-(x,-y) ;xEF,yEF1 defines an isometry a E 0(E) with a2 = 1.

12 that the sign of the determinant is unchanged if we 0 replace e and f by an orthonorinal basis for the plane they span. 2 Let (F,Q) be a positive quadratic form. e. = 0 for all e E F. We shall investigate the orthogonal group O(F), in particular the subgroup generated by reflections. 1. 3 Let Q be a positive quadratic form on the m-dimensional real vector space F. An isometry o E O(F) which is the identity on F written as a product of at most m reflections. 5 PARABOLIC FORMS 17 Proof We proceed by induction on m.

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