Transwedge products
The transwedge product is a generalization of the exterior product and interior product that also includes a transitional sequence of liminal products between exterior and interior.
Transwedge Product
The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as
- $$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.
The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,
- $$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.
When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$.
If $$k > \operatorname{gr}(\mathbf a)$$ or $$k > \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.
An equivalent definition for the transwedge product is given by
- $$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,
where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.
For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.
The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,
- $$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,
where $$n$$ is the dimension of the algebra.
The geometric product of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)
For vectors $$\mathbf a$$ and $$\mathbf b$$, we have
- $$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,
where the dot product on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.
For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have
- $$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,
where the dot product is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.
Transwedge Antiproduct
The transwedge antiproduct of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as
- $$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,
where the wedge and antiwedge products have traded places, the bulk dual has become the weight dual, and we are now summing over all basis elements of antigrade $$k$$.
The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,
- $$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,
The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.
