MeatAxe
2.4
Programs for working with modular representations

pwkond Options [Gnptbk] [i List] [e List] Name [Name ...]
After the irreducible constituents of a module, or a number of modules, have been found with chop, this program can be used
When more than one module is specified on the command line, the peak words found by pwkond are "global", i.e., each peak word selects exactly one of the constituents of alle the modules. Running pwkond successively on two modules does not generally produce global peak words, since a peak word found for module M may have a nonzero nullity on a different constituent that occurs in another module N but not in M.
The e option can be used to exclude certain words from the search. List is a list of integers or ranges of integers, for example "e 57,82112,289". Using "i" you can specify a list of words which will be tested first. This can significantly reduce computation time if you already know one or more peak words for a given module. The "n" option disables the condensation phase. If this option is used, the program stops after the peak words have been found. If the "t" option is specified, pwkond transforms the generators of all irreducible constituents to the standard basis defined by the peak word.
For each composition factor there are several output files. If, for example, one composition factor is X10a, pwkond will produce the following files:
If the module is semisimple, pwkond can calculate a basis that respects the decomposition into irreducible constituents. With respect to this basis, the generators are in block diagonal form, where the blocks occur in the order determined by chop. All blocks corresponding to the same constituent are equal, not only equivalent, and the blocks occur in their "natural" order (as defined by chop). This is essential for the tensor condensation procedure (see precond). To calculate the semisimplicity basis, use the "b" option. The basis is written to Name.ssb. Using "b" with a module that is not semisimple produces undefined results. Most probably, pwkond will stop with the error message "row index out of range", or it will write a singular matrix to Name.ssb.
Internally, a peak word is represented by a pair (n,p) where n is the canonical number of the word (See zmw), and p is a polynomial. The peak word represented by this pair is p(Wn), Wn being the nth word. Without "p", pwkond considers only linear polynomials. If the "p" option is used, pwkond can find polynomials of any degree.
Whenever a peak word is found, the generalized condensation is calculated as follows: The peakword is caculated as a matrix acting on V, which is then repeatedly raised to higher powers until the nullity stabilizes. The stable nullity equals the multiplicity k of the constituent times the degree [E:F] of the splitting field extension. Having a power w^N of the peakword with stable nullity, the condensation onto its kernel, i.e., the projection of V onto V/w^N(V), is determined in the same way as in the zqt program.