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 copies. We can divide the permutations in into two collections: those which can be written as a series of swaps starting with and those which can’t. And throwing on at the beginning of the list just exchanges these two groups for each other in a bijection. If you don’t see that immediately, consider how it’s really the exact same thing as how we solved the airplane seat problem almost two years ago, and Susan’s related Thanksgiving seating problem. Then each permutation in one collection is paired with one in the other collection, and the two in a pair differ by only one swap. This means that one of them will get sent to in the signum representation, and one will get sent to . Since the permuted tensors they give are the same, the two terms will exactly cancel each other out, and we’ll just end up adding together a bunch of zeroes. Neat!
Just like for the symmetric tensors, many different basic tensors will antisymmetrize to the same basic antisymmetric tensor. Again, we use the unique permutation which puts the tensorands in order, but now we use the fact above: if any basis vector of is repeated, then the antisymmetrization is automatically zero. Thus we throw out all these vectors and only consider those -tuples with . This makes it really easy to count the dimension of , because this is just the number of subsets of cardinality of a set of cardinality .
As an example, let’s antisymmetrize . We write out a sum of all the permutations, with signs as appropriate:
We have no repeated terms here because if we did they’d pair off with opposite signs and cancel out. Now you might notice that we didn’t start with the tensorands in order. If we put them in order, we have to use an odd permutation to get . Then we’ll get the same result, but with the opposite signs:
It’s a different result, yes, but it serves just as well as a basic antisymmetric tensor.