MeatAxe: mkhom - Homomorphisms

MeatAxe 2.4

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mkhom - Homomorphisms

## Command Line

    mkhom [Options] [-ts] [-r|-l] [-b Mode] [-H Dim] M N Hom

Options
  Standard options, see Standard Command Line Options

-t
  Calculate generators for M in spinning basis.

-s
  When M equals N, calculate endomorphisms in spinning basis.

-r
  When M equals N, find a generating set of End(M), and calculate the right regular representation.

-l
  Like -r but calculate the left regular representation.

-b Mode
  Save memory. Mode can be 0, 1, or 2 (see description).

-H Dim
  If the radical is given, Dim is the dimension of the head.

M
  First representation.

N
  Second representation.

Hom
  Homomorphisms.

## Input Files

M.cfinfo, N.cfinfo
  Constituent information.

M.1, M.2,...
  Generators in the first representation.

N.1, N.2,...
  Generators in the second representation.

M.rad
  Generators for the head of M (with -H).

MCf.k
  Uncondense matrix produced by pwkond.

## Output Files

M.std
  The spinning basis for M.

Hom.1, Hom.2,...
  A k-basis of Hom(M,N).

M.std.1, M.std.2,...
  Generators in spinning basis (with -t).

## Description

This program calculates a basis for the vector space of homomorphisms between two kG-modules, HomkG(M,N). In the case M=N the program optionally finds a generating set for the algebra of endomoprhisms, EndkG(M), and calculates the corresponding left or right regular representation.

If used without any options, `mkhom` writes the spinning basis of M to M.std, and a k-basis of the homomorphism space to Hom.1, Hom.2,... The latter are given with respect to the spinning basis of M and the original basis of N. To get the homomorphisms with respect to the original bases of M and N, multiply the matrices from the left with the inverse of Hom.

`mkhom` uses peak words of the first module. Thus, before using the program, chop and pwkond must have been run on the first module.

## Implementation Details

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