# Symmetric

Symmetric[{s1,,sn}]

represents the symmetry of a tensor that is symmetric in the slots si.

# Details • The slots si must be different positive numbers. The order of the list is irrelevant.
• Symmetric[{}] and Symmetric[{s}] are both equivalent to the identity symmetry.
• represents the symmetry of a tensor that is symmetric in all its slots.
• If an array is symmetric in a set of slots, then all those slots have the same dimension.

# Examples

open allclose all

## Basic Examples(2)

This array is symmetric:

Declare a rank-4 array to be symmetric in three slots:

Then any transposition involving those slots is equivalent to the original tensor:

## Scope(3)

Symmetry in all slots of a symbolic array:

It can also be specified as follows:

Symmetry in the given slots of a symbolic array:

Symmetric[{}] and Symmetric[{s}] are representations of the absence of symmetry:

Such cases are canonicalized to an empty list of generators:

## Applications(3)

Specify the symmetry of a symmetrized array:

Specify the symmetry of a symbolic array:

Symmetrize several slots of an array:

## Properties & Relations(3)

Detect symmetric matrices:

A symmetric tensor can also be specified by providing explicit generators with phase :

The Wolfram Language automatically detects the equivalence:

Overlapping sets of symmetric slots give full symmetry over all those slots:

Non-overlapping sets do not give full symmetry. The resulting symmetry is described using generators:

## Possible Issues(1)

Dimensions must coincide in all symmetry slots: 