Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Young Tableaux and Tensor Symmetries
The symmetric group acts on a rank- tensor by permuting its slots, and the resulting representation splits into a direct sum of irreducibles labelled by Young tableaux. The subpackage exposes the combinatorial layer (tableaux, hook lengths, dimensions) together with the projectors that decompose a tensor into its irreducible pieces. This tutorial walks through the data structure, the hook-length formula, and the Young projector on small explicit examples.
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Needs["Wolfram`TensorNetworks`"]
In[3]:=
Needs["Wolfram`TensorNetworks`Symmetry`"]
1. Young Tableaux and Symmetry Classes
A Young diagram of size is a partition of into rows of non-increasing length, drawn as a left-justified array of boxes. Filling those boxes with the slot indices gives a standard Young tableau, which the paclet stores as . Every standard tableau labels an irreducible representation of acting on rank- tensors.
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Two extremes are familiar. The single-row shape corresponds to the fully symmetric tensors, the single-column shape to the fully antisymmetric ones, and the mixed shapes label all the intermediate irreps. The constructor accepts either an explicit filling or just a partition (auto-filled left-to-right, top-to-bottom).
{n}
{1,1,…,1}
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yt=
[{3,2}]
 | YoungTableau |
Out[4]=
YoungTableau
 |
|
The shape , total number of boxes , row and column slot lists / , and validity predicate read off directly from the data structure.
In[5]:=
[yt],
[yt],
[yt]
 | TableauShape |
| TableauSize |
| YoungTableauQ |
Out[5]=
{{3,2},5,True}
2. The Hook-Length Formula
The hook of a cell is the set of cells lying to its right in row together with the cells below it in column , plus the cell itself. Frame and James' hook-length formula expresses the dimension of the corresponding irrep as
(i,j)
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n!
∏
(i,j)
where is the hook length at cell . returns all hook lengths arranged in the tableau shape, and picks out a single cell.
h(i,j)
(i,j)
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[{3,2}],
[yt,{1,1}]
| HookLengths |
| HookLength |
Out[6]=
{{{4,3,1},{2,1}},4}
Multiplying the hook lengths and dividing into gives =5, which is what reports. Internally the paclet does not loop over the cells; it uses the Frobenius determinant form, which is in the number of rows rather than in the box count. The reciprocal of the hook product is exposed directly as , with the identity .
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TableauDimension[λ]=n!·HookFactor[λ]
Two partition-level utilities round out the combinatorics. tests whether a list of integers is a partition (positive and non-increasing), and conjugates a Young diagram by swapping its rows and columns. The dimensions of conjugate partitions agree, since transposing a diagram turns the symmetrizer into the antisymmetrizer and vice versa.
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[yt],
[{3,2}],
[{3,2}],
[{2,3}],
[{4,2,1}]
| TableauDimension |
| HookFactor |
| PartitionQ |
| PartitionQ |
| TransposePartition |
Out[7]=
5,,True,False,{3,2,1,1}
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3. Projecting Tensors
The Young symmetrizer associated with a tableau is the algebra element =·, where symmetrizes the slots in each row of and antisymmetrizes the slots in each column. The paclet applies them in exactly that order – rows first, then columns – which is the order that leaves the final column antisymmetry intact. evaluates [T] on a tensor whose rank equals the tableau size.
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T
The unnormalized symmetrizer satisfies =, so the genuine projector onto the irrep is =. This is what applies, with the normalization chosen so that =P.
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The rank-2 case is the most familiar sanity check: shape projects onto the symmetric part (T+), and shape onto the antisymmetric part (T-). A symbolic tensor confirms both.
{2}
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2
T
{1,1}
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T
2×2
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T={{a,b},{c,d}};
T,
[{2}],
T,
[{1,1}]
| YoungProject |
 | YoungTableau |
| YoungProject |
| YoungTableau |
Out[9]=
a,,,d,0,,(-b+c),0
b+c
2
b+c
2
b-c
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1
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At rank three the new feature is the mixed-symmetry irrep , whose representation is two-dimensional. A rank-3 tensor decomposes into the symmetric piece (shape , dimension 1), the antisymmetric piece (shape , dimension 1), and two copies of the mixed irrep (one for each standard filling of , giving total dimension 2). Projecting twice with should be the same as projecting once.
{2,1}
{3}
{1,1,1}
{2,1}
YoungProject[⋯,YoungTableau[{2,1}]]
In[10]:=
SeedRandom[42];R=Round[10RandomReal[1,{2,2,2}],0.01];yt21=
[{2,1}];P1=
[R,yt21];P2=
[P1,yt21];Max[Abs[Flatten[P1-P2]]]
| YoungTableau |
| YoungProject |
| YoungProject |
Out[15]=
0.
The residual is numerically zero, confirming =P. Combining the projectors for all shapes of a given size recovers the original tensor, which is the statement that acts completely reducibly on rank- tensors. The combinatorial bookkeeping that controls when each shape contributes – tableau dimensions and hook lengths – is exactly what the earlier sections compute.
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