Exterior products

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The exterior product is the fundamental product of Grassmann Algebra, which is part of geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct.

Exterior Product

The exterior product is widely known as the wedge product because it is written with an upward pointing wedge. The exterior product between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \wedge \mathbf b$$ and read "$$\mathbf a$$ wedge $$\mathbf b$$". Grassmann called this the progressive combinatorial product.

The defining characteristic of the wedge product is that multiplying any vector $$\mathbf v$$ by itself produces zero: $$\mathbf v \wedge \mathbf v = 0$$. This implies that the wedge product is anticommutative for vectors, so we always have

$$\mathbf v \wedge \mathbf w = -\mathbf w \wedge \mathbf v$$

for vectors $$\mathbf v$$ and $$\mathbf w$$. The wedge product is not anticommutative in general, however. For general basis elements $$\mathbf a$$ and $$\mathbf b$$, reversing the order of the operands satisfies the relationship

$$\mathbf a \wedge \mathbf b = (-1)^{\operatorname{gr}(\mathbf a)\operatorname{gr}(\mathbf b)} \mathbf b \wedge \mathbf a$$ .

The wedge product adds the grades of its operands, so we have

$$\operatorname{gr}(\mathbf a \wedge \mathbf b) = \operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b)$$ .

The following Cayley table shows the exterior products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.

a   b $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
$$\mathbf 1$$ $$0$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
$$\mathbf e_1$$ $$\mathbf e_1$$ $$0$$ $$\mathbf e_{12}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{41}$$ $$-\mathbf e_{321}$$ $$0$$ $$0$$ $$\mathbf e_{314}$$ $$-\mathbf e_{124}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\large\unicode{x1D7D9}$$ $$0$$
$$\mathbf e_2$$ $$\mathbf e_2$$ $$-\mathbf e_{12}$$ $$0$$ $$\mathbf e_{23}$$ $$-\mathbf e_{42}$$ $$0$$ $$-\mathbf e_{321}$$ $$0$$ $$-\mathbf e_{234}$$ $$0$$ $$\mathbf e_{124}$$ $$0$$ $$0$$ $$\large\unicode{x1D7D9}$$ $$0$$ $$0$$
$$\mathbf e_3$$ $$\mathbf e_3$$ $$\mathbf e_{31}$$ $$-\mathbf e_{23}$$ $$0$$ $$-\mathbf e_{43}$$ $$0$$ $$0$$ $$-\mathbf e_{321}$$ $$0$$ $$\mathbf e_{234}$$ $$-\mathbf e_{314}$$ $$0$$ $$\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_4$$ $$\mathbf e_4$$ $$\mathbf e_{41}$$ $$\mathbf e_{42}$$ $$\mathbf e_{43}$$ $$0$$ $$\mathbf e_{234}$$ $$\mathbf e_{314}$$ $$\mathbf e_{124}$$ $$0$$ $$0$$ $$0$$ $$\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{23}$$ $$\mathbf e_{23}$$ $$-\mathbf e_{321}$$ $$0$$ $$0$$ $$\mathbf e_{234}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{31}$$ $$\mathbf e_{31}$$ $$0$$ $$-\mathbf e_{321}$$ $$0$$ $$\mathbf e_{314}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{12}$$ $$\mathbf e_{12}$$ $$0$$ $$0$$ $$-\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$0$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{43}$$ $$\mathbf e_{43}$$ $$\mathbf e_{314}$$ $$-\mathbf e_{234}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{42}$$ $$\mathbf e_{42}$$ $$-\mathbf e_{124}$$ $$0$$ $$\mathbf e_{234}$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{41}$$ $$\mathbf e_{41}$$ $$0$$ $$\mathbf e_{124}$$ $$-\mathbf e_{314}$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{321}$$ $$\mathbf e_{321}$$ $$0$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{124}$$ $$\mathbf e_{124}$$ $$0$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{314}$$ $$\mathbf e_{314}$$ $$0$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{234}$$ $$\mathbf e_{234}$$ $$-\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\large\unicode{x1D7D9}$$ $$\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$

Exterior Antiproduct

The exterior antiproduct is a dual to the exterior product. It is written with a downward pointing wedge and thus called the antiwedge product. The exterior antiproduct $$\mathbf a \vee \mathbf b$$ and read "$$\mathbf a$$ antiwedge $$\mathbf b$$". Grassmann called this the regressive combinatorial product.

Whereas the wedge product combines the full dimensions of its operands, the antiwedge product combines the empty dimensions of its operands. The antiwedge product adds the antigrades of its operands, so we have

$$\operatorname{ag}(\mathbf a \vee \mathbf b) = \operatorname{ag}(\mathbf a) + \operatorname{ag}(\mathbf b)$$ .

The following Cayley table shows the exterior antiproducts between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.

a   b $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
$$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$
$$\mathbf e_1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$\mathbf e_1$$
$$\mathbf e_2$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$\mathbf e_2$$
$$\mathbf e_3$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$\mathbf e_3$$
$$\mathbf e_4$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_4$$
$$\mathbf e_{23}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$-\mathbf e_2$$ $$\mathbf e_3$$ $$0$$ $$\mathbf e_{23}$$
$$\mathbf e_{31}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$-\mathbf e_3$$ $$\mathbf e_{31}$$
$$\mathbf e_{12}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_{12}$$
$$\mathbf e_{43}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_3$$ $$-\mathbf e_4$$ $$0$$ $$0$$ $$\mathbf e_{43}$$
$$\mathbf e_{42}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_2$$ $$0$$ $$-\mathbf e_4$$ $$0$$ $$\mathbf e_{42}$$
$$\mathbf e_{41}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$0$$ $$-\mathbf e_4$$ $$\mathbf e_{41}$$
$$\mathbf e_{321}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_3$$ $$\mathbf e_2$$ $$\mathbf e_1$$ $$0$$ $$-\mathbf e_{12}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{23}$$ $$\mathbf e_{321}$$
$$\mathbf e_{124}$$ $$0$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$-\mathbf e_2$$ $$\mathbf e_1$$ $$0$$ $$-\mathbf e_4$$ $$0$$ $$0$$ $$\mathbf e_{12}$$ $$0$$ $$\mathbf e_{41}$$ $$-\mathbf e_{42}$$ $$\mathbf e_{124}$$
$$\mathbf e_{314}$$ $$0$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$\mathbf e_3$$ $$0$$ $$-\mathbf e_1$$ $$0$$ $$-\mathbf e_4$$ $$0$$ $$\mathbf e_{31}$$ $$-\mathbf e_{41}$$ $$0$$ $$\mathbf e_{43}$$ $$\mathbf e_{314}$$
$$\mathbf e_{234}$$ $$0$$ $$-\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_3$$ $$\mathbf e_2$$ $$0$$ $$0$$ $$-\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{42}$$ $$-\mathbf e_{43}$$ $$0$$ $$\mathbf e_{234}$$
$$\large\unicode{x1D7D9}$$ $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$0$$

De Morgan Laws

There are many possible exterior products and antiproducts. The signs of the results they produce differ in grade-dependent ways, but are otherwise equivalent. The relationship between the product and antiproduct is fixed by a specific choice of dualization function that exchanges full and empty dimensions. We choose the left and right complements as the dualization function and its inverse. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

$$\overline{\mathbf a \wedge \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \vee \overline{\mathbf b}$$
$$\overline{\mathbf a \vee \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \wedge \overline{\mathbf b}$$
$$\underline{\mathbf a \wedge \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \vee \underline{\mathbf b}$$
$$\underline{\mathbf a \vee \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \wedge \underline{\mathbf b}$$

General Properties

The following table lists several general properties of the wedge product and antiwedge product.

Property Description
$$\mathbf a \wedge \mathbf b = -\mathbf b \wedge \mathbf a$$ Anticommutativity of the wedge product for vectors $$\mathbf a$$ and $$\mathbf b$$.
$$\mathbf a \vee \mathbf b = -\mathbf b \vee \mathbf a$$ Anticommutativity of the antiwedge product for antivectors $$\mathbf a$$ and $$\mathbf b$$.
$$(\mathbf a \wedge \mathbf b) \wedge \mathbf c = \mathbf a \wedge (\mathbf b \wedge \mathbf c)$$ Associative law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
$$(\mathbf a \vee \mathbf b) \vee \mathbf c = \mathbf a \vee (\mathbf b \vee \mathbf c)$$ Associative law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
$$\mathbf a \wedge (\mathbf b + \mathbf c) = \mathbf a \wedge \mathbf b + \mathbf a \wedge \mathbf c$$ Distributive law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
$$\mathbf a \vee (\mathbf b + \mathbf c) = \mathbf a \vee \mathbf b + \mathbf a \vee \mathbf c$$ Distributive law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.
$$(t\mathbf a) \wedge \mathbf b = \mathbf a \wedge (t\mathbf b) = t(\mathbf a \wedge \mathbf b)$$ Scalar factorization of the wedge product.
$$(t\mathbf a) \vee \mathbf b = \mathbf a \vee (t\mathbf b) = t(\mathbf a \vee \mathbf b)$$ Scalar factorization of the antiwedge product.
$$s \wedge \mathbf a = \mathbf a \wedge s = s\mathbf a$$ Wedge product of a scalar $$s$$ and any basis element $$\mathbf a$$ that is not a scalar.
$$s \vee \mathbf a = \mathbf a \vee s = s\mathbf a$$ Antiwedge product of an antiscalar $$s$$ and any basis element $$\mathbf a$$ that is not an antiscalar.
$$s \wedge t = 0$$ Wedge product of scalars $$s$$ and $$t$$.
$$s \vee t = 0$$ Antiwedge product of antiscalars $$s$$ and $$t$$.

See Also