MeatAxe: pwkond - Peak Word Condensation

MeatAxe 2.4

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pwkond - Peak Word Condensation

## Command Line

    pwkond Options [-Gnptbk] [-i List] [-e List] Name [Name ...]

Options
  Standard options, see Standard Command Line Options

-G
  Produce output in GAP format.

-n
  Find peak words only, do not condense.

-p
  Use full polynomials instead of linear factors only in peak word search.

-t
  Transform generators into standard basis.

-b
  Calculate a semisimplicity basis.

-k
  Compute kernel of peak words.

-i List
  Word to try first, for example "-i 100,20-35".

-e List
  Words to be excluded, for example "-e 3,20-99".

Name
  Name of the representation.

## Input Files

Name.cfinfo
  Constituent info file.

Name.1, Name.2, ...
  Generators.

NameCF.1, NameCF.2, ...
  Generators on the irreducible constituents. NameCF is the representation name followed by the dimension and a letter to distinguish inequivalent constituents of the same dimension (see chop).

## Output Files

Name.cfinfo
  Constituent info file.

NameCF.1k, NameCF.2k, ...
  Condensed generators on the irreducible constituents.

NameCF.1.std, NameCF.2.std, ...
  Condensed generators in standard basis (with -t).

NameCF.op
  Spin-up script for the standard basis (with -t).

NameCF.np
  Condensed peak word.

NameCF.im
  Image used for the condensation.

NameCF.k
  Peak word kernel (with -k or without -n).

NameCF.ssb
  Semisimplicity basis (with -b).

## Description

After the irreducible constituents of a module, or a number of modules, have been found with chop, this program can be used

- to calulate peak words for the constituents,
- to condense the module using the peak words,
- to transform the generators on the constituents to the standard basis as defined by the peak word kernel, and
- to calculate a basis reflecting the direct decomposition of the module, if the module is semisimple. By definition, a "peak word" for the i-th constituent is an algebra element which has minimal nullity on the i-th constituent and which operates regularly (i.e., with nullity 0) on the other constituents. Als for identifying words (see chop), the nullity of a peak word on its constituent is equal to the degree of the splitting field for that constituent.

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 non-zero 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 57,82-112,289". 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:

- X10a.std.1 and X10a.std.1 are the operation of the generators on the constituent with respect to the standard basis defined by the peak word. These files are created only if the `-t' option is used.
- X10a.op Spin-up script for the standard basis. See ZSB for details.
- X10a.1k and X10a.2k are th action of the generators on the condensed module.
- X10a.np Condensed peak word. This is a nilpotent matrix.
- X10a.im Image of the peak word.
- X10a.k Kernel of the peak word. The .cfinfo file is written each time a peak word is found. So, if the program does not terminate or dies unexpectedly the information about the peak words found so far are not lost.

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.

## Implementation Details

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 n-th 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.