Introduction

This chapter describes Magma functions for computing with groups of Lie type. It was implemented by Scott Murray and Don Taylor based on [CMT].

Given a crystallographic root datum and a ring, we can construct an (untwisted) group of Lie type. Such groups include Lie groups (when the ring in R or C), algebraic groups (when the ring is an algebraically closed field), and finite groups of Lie type (when the ring is a finite field). We plan to implement twisted groups of Lie type in a future release.

Subsections

• The Steinberg presentation
• Bruhat normalisation

The Steinberg presentation

Our approach to computation in groups of Lie type is based on the Steinberg presentation [Ste62]. Let G be the group of Lie type with root datum RD over the ring R. Suppose the roots of RD are alpha_1, ..., alpha_(2N) ordered as in Section Roots, coroots and weights and n is the rank of RD. Then G contains root elements x_r(t)=x_(alpha_r)(t) for t in R, which satisfy the relations:

x_r(t)x_r(u) = x_r(t + u) , and

x_r(t)^(x_s(u)) = x_r(t) prod_(i, j>0) x_(ialpha_r + jalpha_s)(C_(jisr)t^i( - u)^j).

Note that we use the constant C_(jisr) rather than C_(ijrs) because our actions are on the right rather than on the left.

If RD is semisimple, the root elements generate G. In the general case, we have to introduce extra torus elements. Let X=Z^d be the root space of the root datum. Consider the abelian group X tensor R^ x , ie. the set of vectors in R^d with each entry invertible, under entrywise multiplication. Write h_i(t)=alpha_i tensor t. We include the elements of X tensor R^ x as generators of G with relations:

h_r(t) =x_(alpha_r)( - 1)x_(-alpha_r)(1)x_(alpha_r)(t - 1)x_(-alpha_r)( - t^(-1))x_(alpha_r)(t), and

prod_(r=1)^n h_r(t_r) = 1 whenever prod_(r=1)^n t_r^(< alpha_r, beta^star >) = 1 for all beta in Phi.

The Weyl group of G is just the Coxeter group of the root datum RD. We add redundant generators /dot s_r= x_(alpha_r)(1)x_(-alpha_r)( - 1) x_(alpha_r)(1) corresponding to the generators s_r of the Weyl group.

Since our generating set is parametrised by field elements it is not possible to define G within the category GrpFP, so we have created a new category GrpLie.

Bruhat normalisation

The Bruhat decomposition [Car93, Chapter 2] gives us a useful normal form for elements: every g in G can be written in the form uh/dot wu' where

• u is a unipotent element written in the form prod_(r=1)^N x_r(t_r);
• h is a torus element represented as an element of R^d with each entry invertible;
• /dot w = /dot s_(r_1) ... /dot s_(r_k) where s_(r_1) ... s_(r_k) is a reduced word for w in the Weyl group.
• u'=prod_(r in Phi_w^ + )x_r(t_r') where Phi_w^ + = { r | hbox(alpha_r and alpha_rw are positive)} and the terms are in the usual order.

Note that groups of Lie type are designed primarily for fields whose elements are exact. While it is possible to define these groups over real and complex fields (Chapter REAL AND COMPLEX FIELDS), no attempt has been made to control rounding error in this case.