Contraction of multivectors

Question

I am trying to understand a way to contract 1-vectors with p-vectors. So let this be a function $G$ $$G:V\times \wedge^p V\to \wedge^{p-1} V$$ We'd desire the following properties

$$G(\hat{\mathbf{x}}, \hat{\mathbf{x}}\wedge\hat{\mathbf{y}}) = -G(\hat{\mathbf{x}}, \hat{\mathbf{y}}\wedge\hat{\mathbf{x}}) =\pm\hat{\mathbf{y}} $$ $$G(\hat{\mathbf{x}}, \hat{\mathbf{y}}\wedge \hat{\mathbf{z}}) = 0$$

I am wondering how we might elegantly generalize and compute this function. I suppose we could algorithmically calculate $G(\mathbf{u}, \mathbf{v}_1\wedge\dots\wedge\mathbf{v}_p) $ in some manner like this

1. Express $\mathbf{v}_1\dots\mathbf{v}_n$ in a basis including $\mathbf{u}$.
2. Collect terms.
3. Send all terms without a $\mathbf{u}$ to $\mathbf{0}$.
4. Permute and sum to produce a single multivector term with $\mathbf{u}$ first.
5. Remove the leading $\mathbf{u}$ to produce a vector in $ \wedge^{p-1} V$

I am not sure if this is a good or right way to think about it. Any explanation or reference suggestion would be appriciated. Are there elegant ways to express the result in terms of $\mathbf{u}$ and $ \mathbf{v}_1\dots\mathbf{v}_p$?

Answer 1

I would not suggest this algorithm in the general case. It is much easier to use an arbitrary orthogonal (or orthonormal) basis and leverage linearity on the arguments to split the expression into terms only involving factorizable terms on the basis elements.

The problem of contractions in general is well understood, however. The operation naturally arises in use of clifford (or geometric) algebras.