SYMMETRICA-MANUAL -- Characters

Characters

Characters

You have already seen that ordinary character values and character tables of symmetric groups can be evaluated. But there is much more that can be done. For example character tables of alternating groups can be evaluated using

an_tafel().

Moreover ordinary character tables of wreath products (Kranzprodukte in German) of symmetric groups are available, the method used is that of so-called characteristics and the routine is

kranztafel().

Example: Here is a program that evaluates the character table of S_b ≀S_a, by which we mean the wreath product of order a!^bb! (the program can be found in the file ex16.c):
#include"def.h"
#include"macro.h"
main() 
{
OP a,b,c,d,e;
anfang();
a= callocobject(); b= callocobject(); 
c = callocobject(); d=callocobject();
e=callocobject();
scan(INTEGER,a);
scan(INTEGER,b);
kranztafel(a,b,c,d,e);println(c);
println(d);println(e);
freeall(a);freeall(b);
freeall(c);freeall(d);
freeall(e);
ende();
}

Please note that kranztafel() has 5 parameters:

The first two parameters a and b are the degrees of symmetric groups, and the wreath product the table of which is evaluated is

S_b ≀S_a,

of order b!^a⋅ a!.
The third parameter c is the object containing the desired character table.
The fourth parameter d is the vector of orders of conjugacy classes of the wreath product.
The fifth parameter e is the VECTORobject containing the classlabels (which are matrices, the number of columns of which is the degree b while the number of rows is the number of conjugacy classes of S_a.) Please note, that neither the first nor the last label corresponds to the class of the identity element (see the example below)!

For example, if you enter in the above examples the degrees 2 and 3 for a and b, respectively, then the output will look as follows:

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

[2:0:-1:4:0:0:-2:0:1:]

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

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

[0:0:0:4:-2:0:1:1:-2:]

[-2:0:1:4:0:0:-2:0:1:]

[0:0:0:2:0:-2:2:0:2:]

[0:0:0:4:2:0:1:-1:-2:]

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

[6,18,12,1,6,9,4,12,4]

[

[0:0:0:]

[0:0:1:]

[0:0:0:]

[0:1:0:]

[0:0:0:]

[1:0:0:]

[0:0:2:]

[0:0:0:]

[0:1:1:]

[0:0:0:]

[0:2:0:]

[0:0:0:]

[1:0:1:]

[0:0:0:]

[1:1:0:]

[0:0:0:]

[2:0:0:]

[0:0:0:]

]

This shows that in fact we have obtained the character table of S₃ ≀S₂.

If you are interested in modular theory, you can evaluate decomposition matrices and Brauer characters for symmetric groups, using example ex17.c, which is based on the following lines:

scan(INTEGER,n);println(n);
scan(INTEGER,p);println(p);
decp_mat(n,p,b);println(b);
brauer_char(n,p,c);println(c);

The corresponding file is mo.c, and the interested user should carefully note that this particular routine uses a data file called

decommix.dat

which saves all the decomposition and Brauer character tables that were already calculated. Therefore, in order to do a fair play, you better first remove this file decommix.dat, if it is already there, and then you start your modular business from scratch using the above mentioned program.

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

Characters