# tensor2_8.gap

# this file gives GAP code for decomposing the second tensor
# power of the first graded piece of the ring for eight points
# into irreducible modules for the symmetric group S_8.  to
# run it, you can type "gap" at the command line and copy and
# paste the lines of this file which do not begin with a #
# one by one.

# first load and display the character table
s8:=CharacterTable("Symmetric", 8);
Display(s8);

# we know that the first graded piece of the ring is 14
# dimensional.  the table just displayed shows that S_8 has
# exactly two irreducible representations of dimension 14
# (occuring in the 5th and 15th rows of the table).  as these
# differ by a twist by the sign character they have the same
# second tensor power.  since we only care about this tensor
# power, we can use either character.  we take the one in the
# 5th row.
chi:=Irr(s8)[5];

# compute the second tensor power of chi
psi:=chi*chi;

# decompose the second tensor power into irreducibles.  the
# following command will display a list of ordered pairs (d, m),
# one for each irreducible representation of S_8, where d is the
# dimension of the irreducible rep and m is the multiplicity with
# which it occurs in the second tensor power psi.
List(Irr(s8), x->[x[1], ScalarProduct(psi, x)]);

# one can read off directly from the output that psi is
# multiplicity free and that the dimension of the irreducible
# reps which occur in it are 14, 56, 56, 14, 35, 20, 1.  this is
# the fact use in the second paragraph of 2.8.

# to exit gap, type "quit;"