1zbl(1)                           User Commands                          zbl(1)
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NAME

6       zbl - make lower triangular matrix keeping bottom left
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SYNOPSIS

9       zbl [OPTIONS] <Matrix> <Result>
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DESCRIPTION

12       This  program  reads  in  <Matrix>, zeroizes all entries above the main
13       diagonal, and writes out the result.  For example:
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15               Mat                   Result
16       12100201201212212122   10000000000000000000
17       21000000000000001111   21000000000000000000
18       12212120101201201201   12200000000000000000
19       01020010020102012012   01020000000000000000
20       12100012101201012012   12100000000000000000
21       12120212121012012012   12120200000000000000
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23       Notice that the input matrix need not be square, and the output  matrix
24       always has the same dimensions as the input matrix.
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26       If  file  names are omitted, the matrix is read from G1 and output goes
27       to P2.
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29       The purpose of this program is to enable the MeatAxe to check if an ir‐
30       reducible  representation  in  characteristic 2 fixes a quadratic form.
31       This job is not particularly simple — in many ways it is just a  bodge,
32       but it is possible:
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34       1.  Put  the  representation into standard basis using zsp(1) in "stan‐
35           dard basis" mode.
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37       2.  Find the symplectic form fixed by using zsp(1) to make  the  matrix
38           that conjugates the representation to its dual.
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40       3.  Quadratic  forms  can  be represented by lower triangular matrices.
41           Since the representation is in standard basis  (so  all  the  basis
42           vectors  are  images  of  the  first  under the group) the diagonal
43           entries of any fixed quadratic form must all be equal, so try  each
44           field entry in turn by adding that scalar matrix to the bottom half
45           of the symplectic form.
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47       4.  For each quadratic form Q made as in 3, test  if  it  is  fixed  by
48           forming  G^TQG for each generator G, and checking that the diagonal
49           is still the same as it was before (the symplectic form should have
50           been checked before starting).  The check can be done by adding the
51           form to the result, then doing zbl, ztr(1), and  zbl  again  —  the
52           result  will be the zero matrix (use znu(1)) iff the form was fixed
53           (given that the symplectic one was).
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OPTIONS

56       -Q     Quiet, no messages.
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58       -V     Verbose, more messages.
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60       -T <MaxTime>
61              Set CPU time limit
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INPUT FILES

64       Matrix Input matrix
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OUTPUT FILES

67       Result Output matrix.
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SEE ALSO

70       znu(1), zsp(1), ztr(1)
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74MeatAxe                             2.4.24                              zbl(1)
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