Exterior Algebra Notation 1: Multi-Indices
- Wedge Products of Vectors
- Wedge Products in Index Notation, Three Ways
- The Wedge Symbol as Antisymmetrizer?
- Wedge Products in General
Part of a series on Exterior Algebra:
- (This post)
- Star Indices
This is the first of a few posts on Exterior Algebra. It will explore some ways of doing the index notation succinctly and serve as a reference for later posts.
Wedge Products of Vectors
We start by considering the wedge product of
We can use this to write the
We’ll do the same for tensor products, and write:
I will be using the “alternating construction” of the wedge product, which means that
In the first line I’ve sketched the full alternating sum. The second writes this as a sum over signed permutations, and the third with an antisymmetric Levi-Civita symbol and a multi-index
That example indicates my general preference in notation towards extremely succinct expressions. I will tend to omit explicitly stating the range of sums when it’s obvious from the context, and will prefer a Levi-Civita symbol over an explicit sum over permutations. I’ll use Einstein notation, but won’t distinguish the left/right positions of indices because they won’t matter as long as I’m only working with vectors and square (1, 1) matrices. If I want to specify the length of the multi-index in a sum, I will write something like
Wedge Products in Index Notation, Three Ways
Let’s now consider the component representation of a wedge product like
We can expand this in component in three different ways, which I have given names to, as they can otherwise be hard to distinguish. They are:
Tensor Product Basis. The basis of
The wedge product of two vectors in the tensor product basis looks like
Permutation Basis. The basis of
There are two variants of this; the second has antisymmetrized coefficients but therefore has to divide by
Combination Basis, i.e.the basis of distinct
Our example can be written in two equivalent ways:
In total we have three “indexing schemes” to consider: the permutations, the combinations, and the full tensor product. For the case of 2-vectors in 3D the basis elements are:
In general the “tensor product basis” will easiest to write down, but is not necessarily antisymmetric. The permutation basis is manifestly antisymmetric but either has to avoid overcounting or has to have coefficients which are not antisymmetric. The combination basis is the most “natural”, but it can be hard to work out what the component representation ought to be—so my aim here is to arrive at the combination basis expressions in particular, as well as to give examples of my preferred notation.
What are the expansions of
For the tensor product basis, we can either:
- Expand
in components first, then expand the wedge product as tensor products:
- Write
as a tensor product first, and then expand the tensor product in components:
Here I’ve adopted another notation:
These two tensor-basis expression are equivalent; we apparently have a choice as to whether we’d rather antisymmetrize the upper or lower index of
The permutation basis appeared as an intermediate step to the second form of the tensor product:
This expression is the simplest way to represent your typical wedge product. Just write:
This is a sum over all
These expressions appear a lot in books but I find them unappealing.
For the combination basis, we want to write
- starting from the permutation basis with the antisymmetric coefficient, but only summing over one of the
basis elements, thereby removing the overcounting. We get (with the sum over still implied by the Einstein notation):
- or by starting from the tensor basis and gathering up all the terms belonging to the same wedge-basis element
. This merely requires replacing , since the coefficients are already antisymmetric:
In all we have the following equivalent notations:
Note that the combination-basis components:
- are the same as the tensor-basis components, as long as an object is antisymmetric
- are
times the permutation-basis components The first rule is more interesting: it implies that we generally don’t have to think about which basis we’re using. We can in fact write Einstein-notation expressions in the tensor basis and then read them as being in the combination basis, which is what we’ll do.
The 2D examples side-by-side are:
I personally find it easiest to think in the combination basis because it makes the antisymmetry manifest while avoiding any duplicated basis elements. I would generally like to be able to work entirely within it, if not for that that it can be hard to calculate wedge-product formulas without dropping to one of the other bases. The ideal would be able to write antisymmetric tensors in an Einstein notation like:
This would let us do calculations with
Or to write the matrix elements of a wedge power of a matrix as
and
We’ll do all of this, but a question arises: is there any way we can also use the
The Wedge Symbol as Antisymmetrizer?
The idea of using
which gives this expression the appearance of being a “matrix element of
Does this generalize? The cases to consider are:
- The two antisymmetrized permutation-basis expressions:
-
- The two combination-basis expressions
-
In the first of each pairthe
So we could make it work if it only represented a combination-basis sum with antisymmetrization. But I’m still skeptical, because we would not want the same rule to apply—I don’t think—to either the matrix basis elements or the general antisymmetric tensor in
Such a matrix element ought to only have one index antisymmetrized, which can be seen in the case of
This is the determinant, which is usually seen with a permutation
Only one index is antisymmetrized. So the
Instead I’ll continue to use
Sometimes you see this expression with both indices antisymmetrized, which overcounts:
I am hoping to be able to use wedge-indices as components in this way while still using them to represent explicit wedge-products of vectors and unit vectors:
Wedge Products in General
Now we’ll consider the wedge product of two antisymmetric multivectors
Trying to calculate the components of
where in the last line we’ve switched to a sum over all basis elements
This is short enough, but the combinatorial factors are ugly.
The permutation basis would have the same components but divided by
The combination-basis components are as usual the same as the tensor-basis, but we can also try to calculate them directly. The initial expression is simple:
Note the use of
which has three combination-basis sums. The equivalent Einstein-notation expression would just be:
I could see getting used to this, but if that’s a lot we can revert the
This is now equivalent to the tensor-basis coefficient, though we’re calling
As an example, suppose
The first line gives the full tensor-basis expression, which sums over each
Note that while Einstein-notation sums will use
The general case of a 2-vector times a vector will demonstrate the need to deduplicate comb.-basis elements. Take
But if we wanted to write the last line as