zkd Options Field Orbits Perm Kond
The second input file, Perm, must contain one or more permutations. Notice that only the first permutation is read in and condensed. If there are more than one permutation, the others are ignored. Unlike in previous versions of this program, it is not assumed that the orbits are contiguous.
p-part taken has order Nso the user can check it. If this is not the order of the Sylow-p subgroup of the condensation group, the program will not know, so will continue. Normally, however, the condensation subgroup K will have trivial Sylow-p subgroup, or at any rate the Sylow subgroup will have a regular orbit, and in this case at least the condensation is legitimate.
The output is a square matrix with one row and one column for each orbit of K. Abstractly, the condensation can be described as follows. Let G be a permutation group of degree n, F a field of characteristic p and K≤G a p'-subgroup. Then, there is an idempotent
associated to K. Now, let V be a FG-module, for example (as in this program) the natural permutation module , where G acts by permuting the entries of vectors. Then, Ve is an e(FG)e-module, and for any π∈G, the condensed form is eπe, regarded as a linear map on Ve.
To calculate the action of eπe, let be the standard basis such that
for π∈G. A basis of Ve is given by the orbit sums
and with respect to this basis we have
where [m] denotes the orbit containing m.
If K is not a p'-subgroup, e is no longer defined. However, the last formula can still be given a sense by replacing
where m is the highest power of the characteristic which divides any of the orbit sizes. Thus, all but the orbits with maximal p-part are discarded, and the corresponding columns in the output matrix are zero.