Few notations are used in the literature for an Irreducible representation. In most case the dimension or degree of a representation is enough if the symmetry group is known.
Same is true for Young Tableau, Young Tableau are not great for orthogonal groups, because trace need to be excluded after symmetrization.
The highest weight is the most comprehensive notation, and I’ll use it together with the degree. I’ll also illustrate stuff with Sagemath. Sagemath is the public domain symbolic calculator, I run it from web interface, https://cocalc.com/, the free version runs reasonably fast for small formulas.
***** Spatial symmetries of matter and interactions ******
Start with SO(4),
SageMath definition for our future use is: SO4 = WeylCharacterRing(“D2″, style=”coroots”)
The scalar field has highest weight (0,0) and is 1-dimensional. SO4(0,0).degree() = 1
The vector field (1,1) for example vector-potential $A_\mu$ is 4-dimensional. SO4(1,1).degree() = 4
The electro—magnetic field tensor is antisymmetric spatial derivative of $A_mu$. It has 6 elements, and it is also adjoint. SO4(2,0).degree() + SO4(0,2).degree() = 6
The tensor of the gravity is symmetric traceless (2,2), it has 9 components. SO4(2,2).degree() = 9
The space curvature is the Riemann tensor, it has 20 independent components, and it transforms as product of 4 vectors, antisymmetrized on two pairs of indexes, then symmetrized on remaining two. Obtained representtaion is further reducable to SO4(0,0) + SO4(2,2) + SO4(4,0) + SO4(0,4), so 20 is reducable to [1]+[9]+[5]+[5]. Space curvature R is a scalar ~ SO4(0,0), The Ricci tensor transforms similarly to the gravity ~ SO4(2,2), and the Weyl tensor ~ SO4(4,0) + SO4(0,4).
The electron wave function is the bi-spinor field (1,0)+(0,1), it has 4 components, 2 left and 2 right fields, SO4(1,0).degree() + SO4(0,1).degree() = 4
Continue with SO(12),
Sagemath definition is SO12 = WeylCharacterRing(“D6″, style=”coroots”)
The scalar field (0,0,0,0,0,0) is 1-dimensional, SO12(0,0,0,0,0,0).degree() = 1
The Vector is 12-dimensional, SO12(1,0,0,0,0,0).degree = 12
The electro—magnetic tensor, also adjoint is 66 dimensional, SO12(0,1,0,0,0,0).degree() = 66
The Gravity field has 77 components, SO12(2,0,0,0,0,0) = 77
The Riemann tensor has 1716 components, it consists of three pieces: The space curvature scalar SO12(0,0,0,0,0,0) , the 77-dimensional Ricci tensor SO12(2,0,0,0,0,0), and the 1638 dimensional Weyl tensor SO12(0,2,0,0,0,0).degree() = 1638; In other words [1716] = [1]+[77]+[1638].
The bi-spinor field (1,0) + (0,1), has 64 components, SO12(0,0,0,0,1,0).degree() + SO12(0,0,0,0,1).degree() = 64
The group embedding of SO(4) into SO(12) can be done in few ways. Rotations in 66 planes of SO(12) can be organized in 3 groups of 4 axises+6 planes. These are SO(4) subspaces.
## SO(12) -> SO(4)+ SO(4)+ SO(4)
SO12 = WeylCharacterRing(“D6″, style=”coroots”)
SO4x4x4 = WeylCharacterRing(“D2xD2xD2″, style=”coroots”)
Vector becomes sum of vectors:
SO12(1,0,0,0,0,0).branch(SO4x4x4, rule=”orthogonal_sum”)
D2xD2xD2(1,1,0,0,0,0) + D2xD2xD2(0,0,1,1,0,0) + D2xD2xD2(0,0,0,0,1,1)
Gravity becomes some of gravities and sum of products of vectors and product of scalars:
SO12(2,0,0,0,0,0).branch(SO4x4x4, rule=”orthogonal_sum”)
2*D2xD2xD2(0,0,0,0,0,0) + D2xD2xD2(1,1,1,1,0,0) + D2xD2xD2(1,1,0,0,1,1) + D2xD2xD2(2,2,0,0,0,0) + D2xD2xD2(0,0,1,1,1,1) + D2xD2xD2(0,0,2,2,0,0) + D2xD2xD2(0,0,0,0,2,2)
Left Spinor becomes product of spinors with odd number of left spinors:
SO12(0,0,0,0,1,0).branch(SO4x4x4, rule=”orthogonal_sum”)
D2xD2xD2(1,0,1,0,1,0) + D2xD2xD2(1,0,0,1,0,1) + D2xD2xD2(0,1,1,0,0,1) + D2xD2xD2(0,1,0,1,1,0)
Right Spinor becomes product of spinors with even number of left spinors:
SO12(0,0,0,0,0,1).branch(SO4x4x4, rule=”orthogonal_sum”)
D2xD2xD2(1,0,1,0,0,1) + D2xD2xD2(1,0,0,1,1,0) + D2xD2xD2(0,1,1,0,1,0) + D2xD2xD2(0,1,0,1,0,1)
It is possible to allow rotations between 3 subspaces, then it will be so called “tensor” embedding. Accidentally it destroys spinors.
## SO(20) -> SO(5) x SO(4)
SO20 = WeylCharacterRing(“D10″, style=”coroots”)
SO5x4 = WeylCharacterRing(“B2xD2″, style=”coroots”)
SO20(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0).branch(SO5x4, rule=”tensor”)
SO20(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0).branch(SO5x4, rule=”tensor”)
results: B2xD2(1,0,1,1)
B2xD2(0,2,0,0) + B2xD2(0,2,2,2) + B2xD2(2,0,2,0) + B2xD2(2,0,0,2) + B2xD2(0,0,2,0) + B2xD2(0,0,0,2)
***** Internal symmetries of matter and interactions ******
Let’s each component of the vector A_\mu to be also tensor which belong to SU(N) adjoint. The spatial symmetry is preserved to be SO(4) vector. The corresponding field $F_{\mu\nu}$ has same “internal” symmetry as $A_\mu$, and the spatial symmetry as the electro—magnetic field.
Weights for few SU(N) adjoints are below, dimensions are N^2-1
SU3 = WeylCharacterRing(“A2″, style=”coroots”); SU3(1,1).degree() = 8
SU4 = WeylCharacterRing(“A3″, style=”coroots”); SU4(1,0,1).degree() = 15
SU5 = WeylCharacterRing(“A4″, style=”coroots”); SU5(1,0,0,1).degree() = 24
SU6 = WeylCharacterRing(“A5″, style=”coroots”); SU6(1,0,0,0,1).degree() = 35
For material fields one also should define “internal” symmetry, the spatial symmetry is the same as in QED – just bi-spinors. The standard way is to use SU(N) vectors (having N components) for internal symmetry of the material fields. For instance, QCD fields are SU(3) gauge bosons in (1,1)=[8] adjoint and quarks in (1,0)=[3] fundamental representation.
The branching rule of the gauge field (adjoint reps) for the symmetry lowering from SU(5) to SU(3) x SU(2) x U(1) is as follows.
SU5=WeylCharacterRing(“A4″,style=”coroots”)
SU3x2=WeylCharacterRing(“A2xA1″,style=”coroots”)
SU5(1,0,0,1).branch(SU3x2,rule=”levi”) gives A2xA1(0,0,0) + A2xA1(0,1,1) + A2xA1(1,0,1) + A2xA1(1,1,0) + A2xA1(0,0,2) where
A2xA1(0,0,2) is SU(2) adjoint, SU3x2(0,0,2).degree() = 3
A2xA1(1,1,0) is SU(3) adjoint
A2xA1(0,1,1) + A2xA1(1,0,1) are products of vectors
A2xA1(0,0,0) is U(1) scalar
The branching rule of the matter fields (vector reps) is as follows.
For [5^\ast] SU5(0,0,0,1).branch(SU3x2,rule=”levi”) gives A2xA1(0,1,0) + A2xA1(0,0,1)
For [10] SU5(0,1,0,0).branch(SU3x2,rule=”levi”) gives A2xA1(0,0,0) + A2xA1(0,1,0) + A2xA1(1,0,1)
where
A2xA1(0,1,0) is 3-vector (colors) of right d-antiquark
A2xA1(0,0,1) is left neutrino and electron
A2xA1(0,0,0) is right positron
A2xA1(0,1,0) is 3-vector of right u-antiquark
A2xA1(1,0,1) is 6-vector. It contains left 3-color d- and left 3-color u-quarks
Daniel L. Miller Works Collection 