Let’s continue yesterday’s discussion of algebras we can construct from a vector space. Today, we consider the “exterior algebra” on , which consists of the direct sum of all the spaces of antisymmetric tensors
Yes, that’s a capital , not an . This is just standard notation, probably related to the symbol for its multiplication we’ll soon come to.
Again, despite the fact that each is a subspace of the tensor space , this isn’t a subalgebra of , because the tensor product of two antisymmetric tensors may not be antisymmetric itself. Instead, we will take the tensor product of and , and then antisymmetrize it, to give . This will be bilinear, but will it be associative?
Our proof parallels the one we ran through yesterday, writing the symmetric group as the disjoint union of cosets indexed by a set of representatives
and rewriting the symmetrizer in just the right way. But now we’ve got the signs of our permutations to be careful with. Still, let’s dive in with the antisymmetrizers
Where throughout we’ve used the fact that is a representation, and so the signum of the product of two group elements is the product of their signa. We also make the crucial combination of the double sum over into a single sum by noting that each group element shows up exactly times, and each time it shows up with the exact same sign, which lets us factor out from the sum and cancel the normalizing factor.
Now this multiplication is not commutative. Instead, it’s graded-commutative. If and are elements of the exterior algebra, then we find
That is, elements of odd degree anticommute with each other, while elements of even degree commute with everything.
Indeed, given and , we can let be the permutation which moves the last slots to the beginning of the term and the first slots to the end. We can construct by moving each of the last slots one-by-one past the first , taking swaps for each one. That gives a total of swaps, so . Then we write
The dual to the exterior algebra is the algebra of all alternating multilinear functionals on , providing a counterpart to the algebra of polynomial functions on . But where the variables in polynomial functions commute with each other, the basic covectors — analogous to variables reading off components of a vector — anticommute with each other in this algebra.