Artin Groups and Iwahori-Hecke algebras over finite fields

Abstract

In this doctoral thesis, we will determine the image of Artin groups associated to all finite irreducible Coxeter groups inside their associated finite Iwahori-Hecke algebra. This was done in type $A$ by Brunat, Magaard and Marin. The Zariski closure of the image was determined in the generic case by Marin. It is suggested by strong approximation that the results should be similar in the finite case. However, the conditions required to use are much too strong and would only provide a portion of the results. We show in this thesis that they are but that new phenomena arise from the different field factorizations. The techniques used in the finite case are very different from the ones in the generic case. The main arguments come from finite group theory. In high dimension, we will use a theorem by Guralnick-Saxl which uses the classification of finite simple groups to give a condition for subgroups of linear groups to be classical groups in a natural representation. In low dimension, we will mainly use the classification of maximal subgroups of classical groups determined by Bray, Holt and Roney-Dougal for the complicated cases. We find some new $W$-graphs in types $H_4$, $E_6$ and $E_8$ which provide different information from the usual ones. They are all associated in a natural way to a bilinear form which is very complicated to obtain in the previous models. In this model, the bilinear form is obtained using only the two-colorability and its matrix in a well chosen basis is anti-diagonal. The uniqueness properties can probably be extended in a more general setting and understanding which setting this is may be worth considering.