Reverses
Reverses are unary operations in geometric algebra that are analogs of conjugate or transpose operations.
For any element $$\mathbf u$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by
- $$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .
Symmetrically, for any element $$\mathbf u$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the antiwedge product). In general, the antireverse of an element $$\mathbf u$$ is given by
- $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .
The reverse and antireverse of any element $$\mathbf u$$ are related by
- $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,
where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse
- $$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,
and similarly for the antireverse.
The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
In the Book
- Reverses and antireverses are introduced in Section 3.4.