Magma code

On this webpage I collate some code used in my papers. It is grouped according to the paper.

Subspace stabilizers and maximal subgroups of exceptional groups of Lie type

Here we have code to generate the various inequivalent tori that are used to compute the sets in the paper, together with the outputs of those algorithms.

For G₂ we have this which computes the set for L(G) via the maximal-rank subgroup A₂.

For F₄ we have this and this which compute the set for the minimal module via the maximal-rank subgroups A₁⁴ and A₂A₂ respectively, and for L(G) via A₂A₂ we use this code. The outputs of the first two programs can be found here and here.

For E₆ we have this and this which compute the set for the minimal module via the maximal-rank subgroups A₅A₁ and A₂A₂A₂ respectively. The outputs of the programs can be found here and here.

For E₇, because the code is a two-step process as described in the article, and the intermediate file is very large, complete details are available on request. I might be able to put something on here in the future if I can get the file sizes down.

Traces of semisimple elements in exceptional groups of Lie type

Here we collate the sets of traces of semisimple elements in groups of Lie type on various modules, normally the minimal and adjoint for exceptional groups and those involved in those modules for subgroups of exceptional groups.

We use traces for elements of small order, where we use the computer program in Alastair Litterick's PhD thesis. However, this becomes difficult for large element orders, so we compute the conjugacy classes of the normalizer of a torus for a G(q) where n divides q-1, and take their eigenvalues.

For traces the format is as follows: let G be an algebraic group, with a list of modules M₁,M₂,...,M_s for which we want to know the traces of various elements. Each entry X in the sets described below corresponds to a class of semisimple elements of order n, for a positive integer n. Let x denote a representative from a class of such elements. We need to know the traces of x^a for all divisors a of n (except for n itself) on each module in question. Label these divisors d₁,d₂,...,d_r, where d₁=1. The entry X is a list of tuples of traces, where the (i,j)-entry of X is the trace of the element x^d_j on the module M_i.

This is best illustrated by example. Suppose that x has order 4, and has trace 0 on the first module and -3 on the second, and x² has traces -8 and 5 on the two modules. The entry corresponding to x looks like

[ [0,-8], [-3,5] ].

For F₄, we have the following traces available:

Up to 17 and 20, 19, 21, 40, 41, 52.

For F₄, we have the following sets of eigenvalues available:

19, 24, 26, 28, 40, 41, 51, 52, 61, 63, 78.

For E₆, we have the following traces available:

Up to 13, 19.

For E₇, we have the following traces available:

Up to 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26.

For E₇, we have the following sets of eigenvalues available:

16, 19, 20, 21, 24, 25, 26, 28, 31, 33.

I have more than this, but above 33 Magma requires more than one file to store the sets as the size gets too big. If you need any above 33, let me know.

For E₈, we have the following traces available:

Up to 11.

For E₈, we have the following sets of eigenvalues available:

15.