# Tensor Symmetries

Invariance under Phased Permutations | Symmetrization of Arrays |

Symmetry Specifications | More on Symmetry Specifications |

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.

In[2]:= |

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.

## 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[{s_{1},…,s_{n}}] | tensor keeps sign under exchange of any two slots s_{i} |

Antisymmetric[{s_{1},…,s_{n}}] | tensor changes sign under exchange of any two slots s_{i} |

ZeroSymmetric[{s_{1},…,s_{n}}] | symmetry of any zero tensor |

{symgen_{1},…,symgen_{m}} | list of generators of the symmetry of a tensor |

{sym_{1},…,sym_{k}} | direct product of symmetry specifications |

Computation and specification of tensor symmetries.

In[18]:= |

## 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.

### 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.

In[27]:= |

In[32]:= |

The dependent components associated to a given independent component can be obtained using orbit computations under the associated permutation group of the symmetry.

## More on Symmetry Specifications

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

For a phased permutation {perm,ϕ} with ϕ a root of unity, in general you need to have ϕ^{n}1, 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.