The Symmetrizer and Antisymmetrizer Today we’ll introduce two
elements of the group algebra of the symmetric group which have
some interesting properties. Each one, given a representation of
the symmetric group, will give us a subrepresentation of that
representation. The symmetrizer gets its name from the way that
it takes an arbitrary vector and turns it into one that the
symmetric group will act trivially on. Specifically, we use the
element That is, we take all permutations in the symmetric group
and — in a sense — average them out. If we compose this with any
permutation we find But as runs over all the permutations in the
group, does as well. So this is just the symmetrizer back again.
The upshot is that if we have a representation we find that Thus
every vector in the image of is left unchanged by the action of
any permutation. That is, is a subrepresentation on which acts
trivially. The antisymmetrizer, on the other hand, on the other
hand, will give us vectors on which the symmetric group acts by
the