We describe, by example, the conventions used to define a permutation. The list $P\=\left[2\,3\,1\,5\,4\right]$ denotes a permutation acting on a 5 element list $A$ by sending the first element of $A$ to the second slot, the second element of $A$ to the third slot, the third element of $A$to the first slot and so on. Thus, if we apply $P$ to $A\=\left[a\,b\,c\,d\,e\right]$ the result is $\left[c\,a\,b\,e\,d\right]$. The same permutation can be written in cycle notation as $C\=\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]$. As another example the permutation which interchanges $b$ with $c$ in the list $A$ is defined in permutation notation by $P\=\left[1\,3\,2\,4\,5\right]$or in cycle notation as $C\=\left[2\,3\right]$.

A tensor $T$ is a multi-linear map whose arguments are vectors or forms. The command RearrangeIndices defines a new tensor by rearranging the arguments of $T$ according to the permutation $P$. For example, if $T$ is a rank 3 covariant tensor and $S$ = RearrangeIndices(T, [3, 2, 1]) then $S\left(X\,Y\,Z\right)\=T\left(Z\,Y\,X\right)$.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RearrangeIndices(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-RearrangeIndices.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{T1}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{D\_y}\right)$

$\mathrm{T1}≔\mathrm{dx}\mathrm{D\_y}$

$\mathrm{T2}≔\mathrm{evalDG}\left(\left(\left(\mathrm{dx}\&t\mathrm{D\_y}\right)\&t\mathrm{dz}\right)\&t\mathrm{D\_z}\right)$

$\mathrm{T2}≔\mathrm{dx}\mathrm{D\_y}\mathrm{dz}\mathrm{D\_z}$

$\mathrm{RearrangeIndices}\left(\mathrm{T1}\,\left[2\,1\right]\right)$

$\mathrm{D\_y}\mathrm{dx}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[2\,1\,4\,3\right]\right)$

$\mathrm{D\_y}\mathrm{dx}\mathrm{D\_z}\mathrm{dz}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

$\mathrm{D\_y}\mathrm{dx}\mathrm{D\_z}\mathrm{dz}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[1\,3\,4\,2\right]\right)$

$\mathrm{dx}\mathrm{D\_z}\mathrm{D\_y}\mathrm{dz}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[\left[2\,3\,4\right]\right]\right)$

$\mathrm{dx}\mathrm{D\_z}\mathrm{D\_y}\mathrm{dz}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[2\,3\,4\,1\right]\right)$

$\mathrm{D\_z}\mathrm{dx}\mathrm{D\_y}\mathrm{dz}$

$\mathrm{RearrangeIndices}\left(\mathrm{T2}\,\left[\left[1\,2\,3\,4\right]\right]\right)$

$\mathrm{D\_z}\mathrm{dx}\mathrm{D\_y}\mathrm{dz}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\right]\,E\right)\:$

$T≔\mathrm{evalDG}\left(\left(\left(\mathrm{dx}\&t\mathrm{dy}\right)\&t\mathrm{du}\right)\&t\mathrm{dv}\right)$

$T≔\mathrm{dx}\mathrm{dy}\mathrm{du}\mathrm{dv}$

$\mathrm{RearrangeIndices}\left(T\,\left[3\,4\,1\,2\right]\right)$

$\mathrm{du}\mathrm{dv}\mathrm{dx}\mathrm{dy}$