CARAT Introduction / Programs / Orbit

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## Program Orbit

### Usage

    Orbit 'file1' 'file2' [-i] [-r] [-l] [-k] [-t] [-L=n] [-S=n] [-p] [-u] [-g] [-R]

file1: matrix_TYP, contains a matrix X whose orbit is to be calculated
file2: bravais_TYP, contains generators of a group G

### Description

Calulates the orbit of the matrix X in file1 under the group G in file2, where the action is specified by the options. Default option is action by left multiplication.

### Options

    -i     : Use the generators given in file2 and their
             inverses to calculate the orbit.
    -r     : Operate from the right.
    -l     : Operate from the left (default).
    -k     : Operate via conjugation, ie.  x -> g x g^-1
    -L=n   : Calculate at most n elements of the orbit.
             0 means infinity.
    -S=n   : If given as -S or -S=0 a generating set for
             the stabilizer is calculated. If given as
             -S=n at most n matrices of the stabilizer
             are calculated.
             S=-1 means ONLY the stabilizer is calculated.
    -p     : Operate on pairs of the form {M,-M}.
    -u     : Operate on the set of rows of the matrix given
             in file1.
    -f     : Operates on quadratic forms via x -> g^-tr x g^-1
    -g     : Operate on sublattices of Z^n spanned by the columns
             (rows) of the matrices gX (Xg) with g in G. Brakets
             apply if given with the -r option.
    -R     : Give representatives of the G-orbits at the end of
             the output.

### Remarks

WARNING: If the orbit is infinite use option -L!

See also for Is_finite and Order.

### Examples

1. Find the stabilizer of a sublattice in the Bravais group of the unit form F=I6.

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