Reverses

Reverses are unary operations in geometric algebra that are analogs of conjugate or transpose operations.

For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\mathbf e_{234}$$since 432 is an odd permutation of 234. In general, the reverse of an element $$\mathbf x$$ is given by


 * $$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$.

Symmetrically, for any element $$\mathbf x$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf x$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}$$, 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 $$\mathbf x$$ is given by


 * $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$.

The reverse and antireverse of any element $$\mathbf x$$ are related by


 * $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}\,\mathbf{\tilde x}$$.

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 $$\mathcal G_{3,0,1}$$.