Algebra Seminar

By a root system, we mean a finite set of vertors in a euclidean space which is stable under the reflections with respect to these vectors. In this talk, we first consider the irreducible root systems R of the classical types. We introduce two arrangements of positive root systems R⁺ and discuss some properties of R⁺by using these arrangements.

Next we consider the affine Weyl groups W_a associated to R. We introduce two of their presentations: alcove forms and permutation forms. We give the transition formulae between these two forms and study some properties of W_a in terms of these forms.