Image of the Artin groups of classical types inside the finite Iwahori–Hecke algebras

Abstract

We determine the image of the Artin groups of types B and D inside the Iwahori-Hecke algebras, when defined over finite fields, in the semisimple case. This generalizes earlier work on type A by Brunat, Magaard and Marin. In this multi-parameter case, this image depends heavily on the parameters.the Artin groups. Since the latter are fundamental groups of algebraic varieties, this also defines interesting finite coverings of these varieties. Since these varieties are defined over Q, this may have applications to inverse Galois theory (see for example [23]). It is also interesting in terms of finite classical groups because we get explicit generators verifying the braid relations for those groups. This can provide interesting constructions of these groups and some of their subgroups by looking at restrictions to parabolic subgroups of the Artin groups.We now introduce various notations which we use throughout the article. For a finite group G, we write O p (G) for its maximal normal p-subgroup and G ′ = [G, G] for its derived subgroup. We write k for the cyclic group of order k, N.H for an extension of N by H which can be split and N : H for a split extension of N by H where in both cases N is the normal subgroup. We write E p n for the elementary abelian group of order p n . If λ is a double-partition of n, we write λ ⊢⊢ n and if T = (T 1 , T 2 ) is a double-tableau associated with λ, we write T ∈ λ and we call T 1 and T 2 the components of T.We write n λ for the number of standard double-tableaux associated with λ. We write I N the identity matrix and E i,j the elementary matrices.Acknowledgments. This article is part of a doctoral thesis directed by O. Brunat and I. Marin. The author thanks I. Marin for a careful reading of the manuscript and suggested improvements. The author thanks O. Brunat and I. Marin for help understanding [3] and [4] and discussions on some proofs of this article. The author also thanks K. Magaard for suggestions which were a great help to simplify proofs for the low dimensional cases. The author thanks R. Chaneb for pointing out [2] for maximal subgroups of low-dimensional finite classical groups. Type BLet p be a prime, n ∈ N ⋆ , α ∈ F p of order a greater than n and not in {1, 2, 3, 4, 5, 6, 8, 10} and β ∈ F p \ {−α i , −(n − 1) ≤ i ≤ n − 1} different from 1. We set F q = F p (α, β). The Artin group of type B is the group generated by the elements T = S 0 , S 1 , . . . , S n−1 verifying the relationThe associated Iwahori-Hecke F q -algebra is defined by the generators indexed in the same way as for the Artin group and verifying the previous relations and deformations of the relations of order 2 of the Coxeter groups : (T − β)(T + 1) = 0 and for i ∈ [[1, n − 1]], (S i − α)(S i + 1) = 0. In the following, we identify the Artin group with its image inside the Iwahori-Hecke algebra. We write ℓ 1 , ℓ 2 for the length functions on A Bn = T, S i i∈ [[1,n−1]] such that for all i ∈ [[1, n − 1]], ℓ 1 (S i ) = 1, ℓ 1 (T ) = 0, ℓ 2 (S i ) = 0 and ℓ 2 (T ) = 1.In Section 2.1 we give the irreducib...