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Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`Symmetry`
YoungSymmetrize  |
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The result equals [, ], where is [] and is []. is the raw Young operator without the normalization that makes idempotent.
n!
d
t
yt
n
yt
d
yt
d/n!
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The rows are symmetrized first (), then the columns are antisymmetrized (), so that the column antisymmetry of the Young symmetrizer = survives in the output.
b
T
a
T
c
T
a
T
b
T
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For a rank-2 and shape , returns (twice the symmetric part); for shape it returns (twice the antisymmetric part). The factor of two is exactly the = 2 normalization missing relative to .
tensor
{2}
t+
{1,1}
t-
n!/d
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The output is unnormalized: applying twice scales the result by an extra factor and is not idempotent. The idempotent operator is ; for explicit irrep projectors use that symbol instead.
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The rank must equal []; mismatched ranks issue and return . Inputs whose second argument is not a valid leave unevaluated.
tensor
tableau
YoungSymmetrize::rank
Examples
(7)
Basic Examples
(1)
Symmetrize a rank-2 tensor over the row of the totally symmetric tableau {2}; the result is the unnormalized symmetric part t+:
T
t
In[1]:=
 | YoungSymmetrize |
 | YoungTableau |
Out[1]=
{{2,5},{5,8}}
Antisymmetrize over the column of the totally antisymmetric tableau {1,1}; the result is the unnormalized antisymmetric part t-:
T
t
In[2]:=
| YoungSymmetrize |
| YoungTableau |
Out[2]=
{{0,-1},{1,0}}
Apply the mixed-symmetry projector of shape {2,1} to a rank-3 symbolic array:
In[3]:=
| YoungSymmetrize |
| YoungTableau |
Out[3]=
{{{0,2a[1,1,2]-a[1,2,1]-a[2,1,1]},{0,a[1,2,2]+a[2,1,2]-2a[2,2,1]}},{{-2a[1,1,2]+a[1,2,1]+a[2,1,1],0},{-a[1,2,2]-a[2,1,2]+2a[2,2,1],0}}}
Scope
(4)
Applications
(1)
Properties & Relations
(1)
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