Let’s continue yesterday’s project by considering antisymmetric
tensors today. Remember that we’re starting with a tensor
representation of on the tensor power , which also carries an
action of the symmetric group by permuting the tensorands. The
antisymmetrizer (for today) is an element of the group algebra ,
and thus defines an intertwiner from to itself. Its image is
thus a subrepresentation of acting on . Now let’s again say has
finite dimension and pick a basis of . Then we again have a
basis of given by -tuples of basis elements of , and the
permutation again acts by So let’s see what the antisymmetrizer
looks like. It’s just like the symmetrizer, except we have a
sign in each term given by the signum representation of the
symmetric group. Immediately this tells us something very
interesting: if the same basis element of shows up twice in the
basic tensor, then the antisymmetrization of that tensor is
automatically zero! This is because there’s some swap exchanging
the two slots where we find the two co