MeatAxe: zbl - Bottom Left

MeatAxe 2.4

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zbl - Bottom Left

## Command Line

    zbl [Options] Input Output

Options
  Standard options, see Standard Command Line Options

Input
  Input matrix

Output
  Result matrix

## Input Files

Input
  Input matrix

## Output Files

Output
  Result matrix

## Description

This program reads in a matrix, zeroizes all entries above the main diagonal, and writes out the result. For example:

            Mat                  Result
    12100201201212212122  10000000000000000000
    21000000000000001111  21000000000000000000
    12212120101201201201  12200000000000000000
    01020010020102012012  01020000000000000000
    12100012101201012012  12100000000000000000
    12120212121012012012  12120200000000000000

Notice that the input matrix need not be square, and the output matrix has always the same dimensions as the input matrix.

If file names are omitted, the matrix is read from G1 and output goes to P2.

The purpose of this program is to enable the MeatAxe to check if an irreducible representation in characteristic 2 fixes a quadratic form. This job is not particularly simple -- in many ways it is just a bodge, but it is possible:

1. Put the representation into standard basis using zsp in "standard basis" mode.
2. Find the symplectic form fixed by using zsp to make the matrix that conjugates the representation to its dual.
3. Quadratic forms can be represented by lower triangular matrices. Since the representation is in standard basis (so all the basis vectors are images of the first under the group) the diagonal entries of any fixed quadratic form must all be equal, so try each field entry in turn by adding that scalar matrix to the bottom half of the symplectic form.
4. For each quadratic form Q made as in 3, test if it is fixed by forming for each generator , and checking that the diagonal is still the same as it was before (the symplectic form should have been checked before starting). The check can be done by adding the form to the result, then doing zbl, ztr and zbl again --- the result will be the zero matrix (use znu) iff the form was fixed (given that the symplectic one was).