Antisymmetric

Antisymmetric[{s1,,sn}]

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

Details

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

Examples

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Basic Examples  (2)

This matrix is antisymmetric:

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

Then any transposition involving those slots is equivalent to the original tensor, perhaps up to a sign:

Scope  (3)

Antisymmetry in all slots of a symbolic array:

It can also be specified as follows:

Antisymmetry in the given slots of a symbolic array:

Antisymmetric[{}] and Antisymmetric[{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 antisymmetric matrices:

An antisymmetric tensor can also be specified by providing explicit generators with phase :

The Wolfram Language automatically detects the equivalence:

Overlapping sets of antisymmetric slots give full antisymmetry over all those slots:

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

Possible Issues  (1)

Dimensions must coincide in all symmetry slots:

Introduced in 2012
 (9.0)