SYMMETRICA-MANUAL -- The arithmetic

The arithmetic

The arithmetic

For the ordinary and the modular representation theory of symmetric groups we need only prime fields to take into account, so that most of the concerning programs use the integers together with modular addition and multiplication only, rational numbers are treated as pairs of integers. But as soon as orthogonal forms of ordinary irreducible representations of symmetric groups or subgroups like alternating, dihedral or cyclic groups are involved, then we have to use extension fields of the rational field. The corresponding routines for radical extensions or cyclotomic fields are due to T. McDonough. Here is an example for the use of squares, it constructs the element 3/2+5/6sqrt(3):

        m_i_i(3L,a);
        m_i_i(2L,b);
        m_i_i(5L,c);
        m_i_i(6L,d);
        m_ou_b(a,b,koeff1);
        m_ou_b(c,d,koeff2);
        squareroot(a,a);
        mult_apply(a,koeff2);
        add(koeff1,koeff2,sqrad);

You see that m_ou_b() makes fractions, the rest is obvious. Such radicals show up in character tables of alternating groups. These character tables can be evaluated using

an_tafel()

Here is the output for A₅:

[1:1:1:1:1:]

[-1:-1:1:0:4:]

[0:0:-1:1:5:]

[ 1/2 ( 1 - sqr(5) ): 1/2 ( 1 + sqr(5) ):0:-1:3:]

[ 1/2 ( 1 + sqr(5) ): 1/2 ( 1 - sqr(5) ):0:-1:3:]

At present the input of elements of such extensions of Q is quite cumbersome. For example, if you want to apply symmetry adapted bases to a case with dihedral symmetry, so that you have to enter an irreducible representation of D₆, say the representating matrix

(-1)/(2) (-sqrt(3))/(2)

(sqrt(3))/(2) (-1)/(2)

then you have to enter

2 2 125

1 4 1 1 1 1 1 2

1 4 1 3 1 -1 1 2

1 4 1 3 1 1 1 2

1 4 1 1 1 1 1 2

We apologize for that! This will be improved in due course.

harald.fripertinger "at" uni-graz.at, May 26, 2011

The arithmetic