elliptic.at

The purpose of this file is listing conjugacy classes of elliptic elements of
the Weyl group for arbitrary root data (the main application will be to
standard Levi subsystems, which can have any type). We shall proceed by
reduction to the simple case, whence we start by handling simple root data.

Elliptic Weyl group elements are those whose reflection action does not have
any eigenvalue 1; what actually matters here is "relatively elliptic"
elements, which have no eigenvalue 1 for the restriction to the span of the
roots (so any nontrivial central torus, which prevents any elements from being
elliptic, is ignored). Every Weyl group element is relatively elliptic for a
Levi subgroup (not necessarily standard) and hence conjugate to an element
relatively elliptic for a standard Levi subgroup; this serves in
conjugacy_classes.at as basis for the generation of representatives of all
conjugacy classes of Weyl group elements.

Our generation is based on a reduction to the simple case (and reconstruction
from elements for the simple factors); then case by case according to type.

Elliptic conjugacy classes in classical groups. This is based on an explicit
construction, for each type (B and C are equal). Implementation will be done
below inside the function |elliptic_simple|, so there is no code here.

In type $A_n$ there is just the Coxeter element, with word |#n=[0,1,...,n-1]|
(in the symmetric group, the conjugacy class of maximal length cycles).

In type $B_n$, $C_n$, using realisation of $W$ as signed permutations of $n$,
elliptic elements are products of cycles with on each cycle an odd number of
sign changes. To generate the classes, we find the partitions of $n$ (from
combinatorics.at), for each partition make a corresponding product of cycles
(one for each part), each composed with a sign flip on the final element
(which flip is obtained by conjugation from the final simple reflection).

For type $D_n$ we follow the same idea, but it is complicated by the fact that
sign flips come in pairs, so we can only use partitions with an even number of
parts, and double sign flips must be conjugated from the product of the two
final simple reflections (those on the teeth of the forked $D_n$ diagram).

Elliptic conjugacy classes in exceptional types. We simply tabulate here.