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Wolfram`TensorNetworks`Symmetry`

YoungTableau

YoungTableau [rows]
	represents a Young tableau, where rows is a list of lists of slot indices.

YoungTableau [partition]
	constructs a standard tableau from an integer partition , auto-filling rows in reading order.

Details and Options

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This symbol lives in the subpackage context

"Wolfram`TensorNetworks`Symmetry`"

. Load it explicitly with

Needs

["Wolfram`TensorNetworks`Symmetry`"]

; loading only the main context does not auto-load this subpackage.

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A valid Young tableau has rows of non-increasing length and distinct positive-integer entries. The convenience check

YoungTableauQ

caches its result via the

System`Private`HoldValidQ

pattern, so validation is paid for once per object.

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For a standard tableau the entries are a permutation of

1,…,n

where

is the total number of boxes; downstream symmetrizers such as

YoungSymmetrize

and

YoungProject

require this convention.

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The partition-form constructor numbers cells left-to-right, top-to-bottom:

YoungTableau

[{3,2}]

produces

YoungTableau

[{{1,2,3},{4,5}}]

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The summary-box icon shows the underlying Young diagram, drawn from the same row data; the displayed properties are the shape (partition), the filling label, and the irrep dimension via the hook-length formula.

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The filling label reads "Canonical" when the stored rows are the row-reading standard Young tableau of the shape (the default produced by

YoungTableau

[partition]

), "Standard" when the rows and columns are strictly increasing but the filling is not the canonical one, and "Explicit" otherwise. The label is content-based, so a hand-typed canonical filling such as

YoungTableau

[{{1,2,3},{4,5},{6}}]

also reads "Canonical".

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Property accessors include

, and

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StandardTableauQ

tests whether the filling is a standard Young tableau, that is, whether rows and columns are strictly increasing; the canonical filling always satisfies this.

Examples

(22)

Basic Examples

(4)

Construct a Young tableau from explicit rows:

In[1]:=