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


  • 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.
  • Symmetric[All] 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.


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:

Introduced in 2012