# Tensor Symmetries

Tensors of rank 2 or higher that arise in applications usually have symmetries under exchange of their slots. For example, the inertia tensor, the stress-energy tensor, or the Ricci curvature tensor are rank-2 fully symmetric tensors; the electromagnetic tensor is a rank-2 antisymmetric tensor; and the Riemann curvature tensor and the stiffness tensor are rank-4 tensors with nontrival symmetries. The Wolfram System has a general language to describe an arbitrary symmetry under permutations of the slots of any tensor and implements efficient algorithms to give those tensors a unique canonical form under those symmetries, an essential step in symbolic tensor computations.

## Invariance under Phased Permutations

The basic action on a tensor is formed by a transposition by a permutation and multiplication by a root of unity. If a tensor is invariant under such action, it can be said that the tensor has symmetry.

 {permutation,phase} general form of a symmetry generator TensorTranspose[tensor,gen] action of a symmetry generator on a tensor

Symmetry generators and tensor transposition.

A symmetry generator, or phased permutation, is a list containing a permutation describing how to permute the slots of a tensor and a root of unity that will simultaneously multiply the tensor:
 In:= Out= This rank-4 array has symmetry because it stays invariant under the action of a phased permutation:
 In:= In:= Out= In:= Out= However, it is not invariant under this other generator:
 In:= Out= In:= Out= Successive application of generators is equivalent to a product of generators, where phases and permutations are multiplied separately. In fact, if a tensor is invariant under two phased permutations, then it will be also invariant under their product. Hence, the set of phased permutations under which a tensor is invariant form a group, the slot symmetry group of the tensor.

If an array is invariant under a generator, then it is also invariant under its powers:
 In:= Out= Out= Out= Out= In:= Out= In:= Out= In:= Out= ## Symmetry Specifications

The function TensorSymmetry returns a complete description of the transposition symmetry of a tensor. It can be given as a named symmetry or as a list of some symmetry generators from which the rest can be constructed by permutation products and powers.

 TensorSymmetry[tensor] find the transposition symmetry of tensor Symmetric[{s1,…,sn}] tensor keeps sign under exchange of any two slots si Antisymmetric[{s1,…,sn}] tensor changes sign under exchange of any two slots si ZeroSymmetric[{s1,…,sn}] symmetry of any zero tensor {symgen1,…,symgenm} list of generators of the symmetry of a tensor {sym1,…,symk} direct product of symmetry specifications

Computation and specification of tensor symmetries.

A symmetric matrix:
 In:= Out= An antisymmetric rank-3 array:
 In:= Out= The absence of symmetry, or identity symmetry, is represented by an empty list of generators:
 In:= Out= Representation of the symmetry of a zero tensor:
 In:= Out= In the general case, TensorSymmetry returns the symmetry of a tensor as a list of generators. Permutations are given in cyclic form:
 In:= In:= Out= In:= Out= ## Symmetrization of Arrays

It is possible to increase the symmetry of an array by symmetrizing it, using the function Symmetrize. The result is given as a structured array of type SymmetrizedArray. For more information on this type of structure see "Symmetrized Arrays".

 Symmetrize[tensor,sym] symmetrize tensor to the symmetry sym SymmetrizedArray[rules,dims,sym] construct an array with symmetry, giving its independent components StructuredArray[SymmetrizedArray,dims,data] structured array representation of an array with symmetry

Tensor symmetrization and symmetrized arrays.

Take a general array with no symmetry:
 In:= Out= Project into its antisymmetric part, with the result given as a structured array of type SymmetrizedArray, which stores only independent components:
 In:= Out= It only contains one independent component:
 In:= Out= That component is repeated multiple times in the normal form of the array:
 In:= Out= It is a projection because further antisymmetrization does not change the array:
 In:= Out= Symmetrization in any pair of levels now yields the zero array, with maximal symmetry:
 In:= Out= ### Independent and Dependent Components

When a tensor or array has symmetry, then there is less freedom to specify its components. The symmetries actually specify relations among the components, and only some of them are independent.

 SymmetrizedIndependentComponents[dims,sym] independent components of an array with given dimensions and symmetry SymmetrizedDependentComponents[comp,sym] dependent components associated to a given component under a symmetry

Independent and dependent components.

Take an array with symmetry:
 In:= In:= Out= In:= Out= The array has only four entries freely specifiable, and you can show that this is actually the maximum number compatible with its dimensions and symmetry:
 In:= Out= In:= Out= There are six independent components for an array of dimensions {3,3,3,3} with the transposition symmetries of a Riemann tensor:
 In:= In:= Out= Asymptotically, it grows as , where is the order of the symmetry group, 8 for the Riemann symmetry:
 In:= Out= In:= Out= The dependent components associated to a given independent component can be obtained using orbit computations under the associated permutation group of the symmetry.

Take a generic symmetrized array of rank 3 in dimension 3:
 In:= Out= These are the 10 independent components, given as positions:
 In:= Out= And these are their respective orbits of dependent components:
 In:= Out= You can check that indeed all components in each orbit coincide:
 In:= Out= ## More on Symmetry Specifications

The function SymmetrizedIndependentComponents is used to give some more examples of symmetry specifications.

Symmetry given by a single generator, with complex phase:
 In:= Out= Symmetry given by several generators:
 In:= Out= Product of symmetries:
 In:= Out= Named symmetries can be mixed with symmetries gives by generators, both using permutation lists or permutation cycles:
 In:= Out= For a phased permutation {perm,ϕ} with ϕ a root of unity, in general you need to have ϕn1, where n is the permutation order of perm, as given by PermutationOrder[perm]. Otherwise, the generator can only be a symmetry of the zero tensor, and in such a case that generator is referred to as inconsistent or self-inconsistent. A tensor symmetry may also be compatible only with the zero tensor, even if it is expressed with generators that are self-consistent. That is, the composition of self-consistent generators may give a self-inconsistent generator.

The generator {{2,3,1},-1} is inconsistent because the order of the permutation is 3, but -1 is a root of unity of order 2. The result is the empty list, because the zero tensor does not have independent components:
 In:=  Out= These two generators are consistent, but their combination gives a symmetry only compatible with the zero tensor:
 In:=  Out= Both cases are equivalent to specifying the zero symmetry directly:
 In:= Out= 