Matrix representations |

as well as three versions of the rational integral form:odg(), sdg()

The first one is the one described in H. Boerner's book, first edition, the second one is described in the book by D. E. Rutherford, in Boerner'st book, second edition, as well as in the book by G. D. James and A. Kerber. The third form is due to W. Specht, and it has the advantage that it allows to evaluate a matrix representation corresponding to a skew diagram. Here is a little program:bdg(),ndg(),specht_dg()

scan(PARTITION,part); scan(PERMUTATION,perm); bdg(part,perm,D); tex(D); sdg(part,perm,D); tex(D); odg(part,perm,D); tex(D); specht_dg(part,perm,D); tex(D);If you input the partition

-1 | -1 | 1 | 1 | 0 |

-1 | 0 | 0 | 0 | 1 |

0 | -1 | 0 | 0 | 0 |

-1 | 0 | 0 | 1 | 0 |

0 | -1 | 0 | 1 | 0 |

1 / 4 | -3 / 8 | 3 / 8 | -9 / 16 | 0 |

-1 / 2 | 1 / -4 | -3 / 4 | 3 / -8 | 0 |

-1 / 2 | -1 / 4 | 1 / 4 | 1 / 8 | -2 / 3 |

1 | 1 / -6 | 1 / -2 | 1 / 12 | 4 / -9 |

0 | 1 | 0 | 1 / -2 | 1 / -3 |

1 / 4 | -1 / 4 sqrt( 3 ) | 1 / 4 sqrt( 3 ) | -3 / 4 | 0 |

-1 / 4 sqrt( 3 ) | 1 / -4 | -3 / 4 | 1 / -4 sqrt( 3 ) | 0 |

-1 / 4 sqrt( 3 ) | -1 / 4 | 1 / 4 | 1 / 12 sqrt( 3 ) | -1 / 3 sqrt( 6 ) |

3 / 4 | 1 / -12 sqrt( 3 ) | 1 / -4 sqrt( 3 ) | 1 / 12 | 1 / -3 sqrt( 2 ) |

0 | 1 / 3 sqrt( 6 ) | 0 | 1 / -3 sqrt( 2 ) | 1 / -3 |

0 | -1 | 0 | 0 | 0 |

0 | 0 | -1 | 0 | 0 |

1 | 0 | -1 | -1 | 1 |

0 | 0 | -1 | 0 | 1 |

0 | 1 | -1 | -1 | 1 |

For the modular case, SYMMETRICA provides the routines used by A. Golembiowski which are based on the definition of the irreducible modules given by M. Clausen using standard bideterminants. The routine is called

There are also routines for the evaluation of the ordinary irreducible polynomial representations of general linear groupsmoddg().

uses symmetry adapted bases in order to decompose the tensor productglmndg()

scan(INTEGER,m); scan(INTEGER,n); glmndg(m,n,M,0L); println(M); for(i=0L;i<S_V_LI(M);++i) glm_homtest(m,S_V_I(M,i)); glmndg(m,n,M,1L); println(M); for(i=0L;i<S_V_LI(M);++i) glm_homtest(m,S_V_I(M,i));(please note that it contains a test for homomorphism property). In case you enter 2 for

*D = 3 * 1(D _{1}) + 1 * 1(D_{2})*

*[*

*[1*

*[2:]*

*:1 *

*[1:1:]*

*:1*

*[0:2:]*

*:]*

*[2*

* vdots *

The first row of this output indicates that the tensor square *Ä ^{2}C^{2}*
decomposes (as

A more general monomial, say()

1·x _{11}^{2}1·x _{11}x_{12}1·x _{12}^{2}2... ... ... ...

*1*

*[[0:1:0:]*

*[0:0:1:]:]*

Another routine, namely

gives, for a partition ofglpdg()

There is also a way of getting the output in LaTeX-readable form (but up to now it works only for the representations of general linear groups). Here is a corresponding test program:

... glmndg(m,n,M,1L); println(M); for(i=0L;i<S_V_LI(M);++i) latex_glm_dar(S_V_I(M,i)); ...In the case when you enter 2 for

[.

x _{1 1}^{2}2x _{1 1}x_{2 1}x _{2 1}^{2}

..

x _{1 1}x_{1 2}x _{1 2}x_{2 1}+x_{1 1}x_{2 2}x _{2 1}x_{2 2}

.]

x _{1 2}^{2}2x _{1 2}x_{2 2}x _{2 2}^{2}

You see that[]

-x _{1 2}x_{2 1}+x_{1 1}x_{2 2}

harald.fripertinger@kfunigraz.ac.at,

last changed: November 19, 2001

Matrix representations |