MeatAxe: decomp - Decompose a Module

MeatAxe 2.4

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decomp - Decompose a Module

## Command Line

    decomp Options [-ta] Module Endo RadBasis

Options
  Standard options, see Standard Command Line Options

-t
  Transform generators.

-a
  Calculate the action on direct summands.

Module
  Name of the module to decompose.

Endo
  Name of the endomorphism ring.

RadBasis
  Basis for radical series of the endomorphism ring.

## Input Files

Module.1, Module.2, ...
  Generators for the module.

Module.cfinfo
  Constituent information.

Endo.1, Endo.2, ...
  A k-Basis of the endomorphism ring.

Endo.gens.1, Endo.gens.2, ...
  A generating system of the endomorphism ring.

Endo.lrr.1, Endo.lrr.2, ...
  Left regular representation of the endomorphism ring.

Endo.lrr.cfinfo
  Constituent information for the left regular representation. At least chop and mksub must have been run.

Endo.lrr.soc
  Basis if the socle of Endo.lrr (make by soc).

## Output Files

Module.dec
  Basis of the module reflecting the decomposition.

Module.dec.1, Module.dec.2, ...
  Basis of the module reflecting the decomposition. Generators in decomp. basis (with -t).

Module.Comp.1, Module.Comp.2, ...
  Generators on the components (with -a).

## Description

This program decomposes a module into its direct summands, using the head of the endomorphism ring. It assumes that the endomorphism ring and its left regular representation has already be calculated using mkhom. It also expects that chop. pwkond, and rad have been run on the left regular representation.

decomp produces three types of output files:

- A basis of the module reflecting the decomposition is written to Module.dec. With respect to this basis, the generators have a block-diagonal structure with the blocks corresponding to the direct summands.
- If you use the "-t" option, decomp calculates the action of the generators with respect to the decomposition basis, and writes it to Module.dec.1, Module.dec.2,...
- If you use the "-a" option, the program also calculates the action of the generators on each direct summand, i.e., the blocks of the matrices above. They are written to Module.Comp.1, Module.Comp.2, and so on, where Comp is the name of the direct summand, containing the isomorphism type (dimension plus one letter) and a number counting isomorphic summands that occur more than once in the decomposition.

## Implementation Details

The algorithm used by this program was developed by Magdolna Szöke [Sz98].