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TensorSymmetry
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TensorSymmetry

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TensorSymmetry[tensor]

gives the symmetry of tensor under permutations of its slots.

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TensorSymmetry[tensor,slots]

gives the symmetry under permutation of the specified list of slots.

Details and Options

  • TensorSymmetry accepts any type of tensor, either symbolic or explicit, including any type of array.
  • A general symmetry is specified by a generating set of pairs {perm,ϕ}, where perm is a permutation of the slots of the tensor, and ϕ is a root of unity. Each pair represents a symmetry of the tensor of the form ϕ TensorTranspose[tensor,perm]==tensor.
  • Some symmetry specifications have names:
  • Symmetric[{s1,…,sn}]full symmetry in the slots si
    Antisymmetric[{s1,…,sn}]antisymmetry in the slots si
    ZeroSymmetric[{s1,…,sn}]symmetry of a zero tensor
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about tensors
    SameTestAutomaticfunction to test equality of expressions
    ToleranceAutomatictolerance for approximate numbers
  • For exact and symbolic arrays, the option SameTest->f indicates that two entries aij… and akl… are taken to be equal if f[aij…,akl…] gives True.
  • For approximate arrays, the option Tolerance->t can be used to indicate that all entries Abs[aij…]≤t are taken to be zero.
  • For array entries Abs[aij…]>t, equality comparison is done except for the last bits, where is $MachineEpsilon for MachinePrecision arrays and for arrays of Precision .

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

A symmetric matrix:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-l1jrsp
Out[1]=1

An antisymmetric matrix:

In[2]:=2
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https://wolfram.com/xid/0rsy8amms-le6d13
Out[2]=2

A symmetric array of rank 3:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-6qgilb
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-82w844
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

Find symmetry in an array:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-dbf8n0
Out[1]=1

Find symmetries in complex arrays:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-u796hm
Out[1]=1

Symmetry of a SymmetrizedArray object:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-wj2tue
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rsy8amms-2tswmd
Out[2]=2

Symmetry of a SparseArray object:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-j2mgw5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rsy8amms-ecq6v7
Out[2]=2

Specify the symmetry of a symbolic array:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-z2l6p6
In[2]:=2
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https://wolfram.com/xid/0rsy8amms-gnqc9d
Out[2]=2

Symmetry of its tensor product with itself. Note the exchange symmetry:

In[3]:=3
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https://wolfram.com/xid/0rsy8amms-4mnhck
Out[3]=3

A fully symmetric rank 3 array:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-glntma
Out[1]=1

Complete symmetry:

In[2]:=2
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https://wolfram.com/xid/0rsy8amms-f2c8kg
Out[2]=2

Symmetry in a subset of slots:

In[3]:=3
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https://wolfram.com/xid/0rsy8amms-1z9c3k
Out[3]=3

Symmetry of an array of zeros:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-fejnuo
Out[1]=1

Options  (3)Common values & functionality for each option

Assumptions  (1)

Specify locally the properties of the tensors:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-lr5mpx
Out[1]=1

With no assumptions, the symmetry is unknown:

In[2]:=2
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https://wolfram.com/xid/0rsy8amms-thdqaj
Out[2]=2

SameTest  (1)

This matrix is symmetric for a positive real , but TensorSymmetry gives no symmetry:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-em2aht
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-ej0bsf
Out[2]=2

Use the option SameTest to get the correct answer:

In[3]:=3
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https://wolfram.com/xid/0rsy8amms-63zf1y
Out[3]=3

Tolerance  (1)

Generate a fully symmetric random array of depth 4:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-u97cw7
In[2]:=2
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https://wolfram.com/xid/0rsy8amms-w74trk
In[3]:=3
✖
https://wolfram.com/xid/0rsy8amms-zj1bl1
Out[3]=3

The addition of a small perturbation breaks the symmetry:

In[4]:=4
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https://wolfram.com/xid/0rsy8amms-7ooegl
In[5]:=5
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https://wolfram.com/xid/0rsy8amms-18r6kg
Out[5]=5

The symmetry can be recovered by allowing some tolerance:

In[6]:=6
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https://wolfram.com/xid/0rsy8amms-e0wkny
Out[6]=6

Properties & Relations  (5)Properties of the function, and connections to other functions

Test whether a matrix is symmetric:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-zxssrt
In[2]:=2
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https://wolfram.com/xid/0rsy8amms-f71jee
Out[2]=2

Find the symmetry of the matrix:

In[3]:=3
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https://wolfram.com/xid/0rsy8amms-clwtjz
Out[3]=3

Test whether a matrix is antisymmetric:

In[1]:=1
✖
https://wolfram.com/xid/0rsy8amms-j9fdyw
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-4zl6l
Out[2]=2

Find the symmetry of the matrix:

In[3]:=3
✖
https://wolfram.com/xid/0rsy8amms-n6yc3x
Out[3]=3

Only a matrix of zeros can be simultaneously symmetric and antisymmetric:

In[1]:=1
✖
https://wolfram.com/xid/0rsy8amms-yueai8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-ssw1g9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rsy8amms-uwf0n4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rsy8amms-fr8txh
Out[4]=4

Generation of special multidimensional symmetric arrays:

In[1]:=1
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https://wolfram.com/xid/0rsy8amms-kgrwe2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-b3k6pu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rsy8amms-k3ejp4
Out[3]=3

With a different radius, there are other symmetries:

In[4]:=4
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https://wolfram.com/xid/0rsy8amms-idgrpv
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0rsy8amms-0in36m
Out[5]=5

The symmetry of Symmetrize[tensor,sym] is at least sym:

In[1]:=1
✖
https://wolfram.com/xid/0rsy8amms-mi1qre
In[2]:=2
✖
https://wolfram.com/xid/0rsy8amms-6zl05b
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rsy8amms-fyhy2p
Out[3]=3

In some cases the result of Symmetrize[tensor,sym] may have more symmetry than sym:

In[4]:=4
✖
https://wolfram.com/xid/0rsy8amms-z4l4ss
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0rsy8amms-fav4xu
Out[5]=5
Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).
✖
Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).

Text

Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).

✖
Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).

CMS

Wolfram Language. 2012. "TensorSymmetry." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/TensorSymmetry.html.

✖
Wolfram Language. 2012. "TensorSymmetry." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/TensorSymmetry.html.

APA

Wolfram Language. (2012). TensorSymmetry. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorSymmetry.html

✖
Wolfram Language. (2012). TensorSymmetry. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorSymmetry.html

BibTeX

@misc{reference.wolfram_2025_tensorsymmetry, author="Wolfram Research", title="{TensorSymmetry}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/TensorSymmetry.html}", note=[Accessed: 19-June-2025 ]}

✖
@misc{reference.wolfram_2025_tensorsymmetry, author="Wolfram Research", title="{TensorSymmetry}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/TensorSymmetry.html}", note=[Accessed: 19-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_tensorsymmetry, organization={Wolfram Research}, title={TensorSymmetry}, year={2017}, url={https://reference.wolfram.com/language/ref/TensorSymmetry.html}, note=[Accessed: 19-June-2025 ]}

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@online{reference.wolfram_2025_tensorsymmetry, organization={Wolfram Research}, title={TensorSymmetry}, year={2017}, url={https://reference.wolfram.com/language/ref/TensorSymmetry.html}, note=[Accessed: 19-June-2025 ]}