Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`Symmetry`
YoungProject  |
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[, ] = [, ], where is [] and is []. The normalization makes the projector idempotent.
t
yt
d
n!
t
yt
n
yt
d
yt
d/n!
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Idempotency: [[, ], ] is equal to [, ]. This is the defining property of a projector and is the source of the factor.
t
yt
yt
t
yt
d/n!
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For a rank-2 , projecting onto yields the symmetric part , and projecting onto yields the antisymmetric part .
tensor
YoungTableau[{2}]
(t+)/2
YoungTableau[{1,1}]
(t-)/2
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Summing the Young projectors over all standard tableaux of every shape of size recovers the identity on rank- tensors. For rank 2 this reduces to , reflecting the Schur-Weyl decomposition of the action.
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n
YoungProject[t,YoungTableau[{2}]]+YoungProject[t,YoungTableau[{1,1}]]=t
S
n
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The rank must equal []; mismatched ranks return (via ). Inputs that are not a valid leave unevaluated.
tensor
tableau
Examples
(7)
Basic Examples
(1)
Project a rank-2 tensor onto the totally symmetric class:
In[1]:=
 | YoungProject |
 | YoungTableau |
Out[1]=
1,,,4
5
2
5
2
Project the same tensor onto the antisymmetric class:
In[2]:=
| YoungProject |
| YoungTableau |
Out[2]=
0,-,,0
1
2
1
2
Projection is idempotent:
In[3]:=
 | YoungProject |
 | YoungProject |
 | YoungTableau |
| YoungTableau |
| YoungProject |
| YoungTableau |
Out[3]=
True
Scope
(4)
Applications
(1)
Properties & Relations
(1)
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