Linearity of Artin groups of finite type

I will talk on joint work with Arjeh Cohen which extends the recent work on linearity of the Braid groups to Artin groups of finite type. A group is called linear if there is a faithful finite dimensional representation. It has only recently been shown by Krammer and Bigelow that the Braid groups are linear. We have been able to generalize many of the arguments to the Artin groups of finite type, namely A_n, B_n, C_n, D_n, G_2, F_4, E_6, E_7, and E_8. The Braid group is the group A_n. This is done by recognizing many of the arguments of Krammer as results on the root systems of the corresponding type. Using these properties, we are able to produce an action of the Artin group on a vector space which has as its basis the positive roots. We are able to show it is faithful and so the groups are linear.