Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`Symmetry`
SchurDimension  |
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The dimension of the Weyl module labelled by a partition is given by the hook-content formula , where the product runs over the cells of the Young diagram of , is the hook length at , and the numerator is the content of that cell shifted by .
par
dim=
∏
(i,j)
d+j-i
h(i,j)
(i,j)
par
h(i,j)
(i,j)
d+j-i
d
▪
s
λ
x
1
x
d
GL(d)
par
▪
Numerically equals the rank of the single-tableau Young projector on , where .
⊗n
()
d
n=Total[par]
▪
▪
returns and emits when given anything other than an integer partition (a list of weakly decreasing positive integers) or a .
SchurDimension::notpar
Examples
(14)
Basic Examples
(1)
Dimension of the Weyl module of shape {2}, the symmetric square of :
GL
3
3
In[1]:=
SchurDimension[{2},3]
Out[1]=
6
The adjoint of SU(3) at shape {2,1}:
In[2]:=
SchurDimension[{2,1},3]
Out[2]=
8
In[3]:=
Factor[SchurDimension[{2,2},n]]
Out[3]=
1
12
2
n
Scope
(7)
Applications
(3)
Properties & Relations
(3)
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