Matchete`

InvariantTensors

InvariantTensors[alg, {rep₁, rep₂, …}]

returns a basis of invariant tensors (CGs) which can contract the indices of the representations (order preserved).

Details and Options

The result is a list of tensors (SparseArray).
The invariant tensors are invariant when contracted with objects in of given representations. In other words, the indices of the produced tensors are {CRep[rep₁], CRep[rep₂], …}.
All representation bases are consistent also with the tensors from StructureConstants and Generators.
The algebra argument must be a valid Alg[class, n] object.
Representations are labeled with a list of integers corresponding to the Dynkin coefficients of the highest weight in the representation.
The following options can be given:

AntisymmetricIndices	{}	restricts the tensors to be antisymmetric in the given indices
Normalization	Default	the normalization of the norm square of the invariant tensors.
SymmetricIndices	{}	restricts the tensors to be symmetric in the given indices

The options given to Symmetricindices and AntisymmetricIndices can be either a list of index numbers or a list of list of index numbers with indices to be symmetrized between.
The default for value for the norm square normalization of the tensors is , where is the dimension of each representation.

Examples

open allclose all

Basic Examples (1)

An invariant tensor between three fundamental representations of SU(3)/A₂ (note the use of SU[3] as short-hand for the algebra):

The only such tensor is proportional to the rank-3 Levi-Civita tensor, though the default normalization differs.

Options (3)

AntisymmetricIndices (1)

In the product of an anti-fundamental, a fundamental, and two adjoint representations of SU(3), there is a single invariant tensor that is antisymmetric in the two adjoint indices. It is proportional to f_cab:

Normalization (1)

The default normalization of the tensor is unusual for the case of the Levi-Civita tensor of SU(3). By setting the normalization, we may obtain the ordinary choice

SymmetricIndices (1)

In the product of an anti-fundamental, a fundamental, and two adjoint representations of SU(3), there is is a two-dimensional space of invariant tensors that are symmetric in the two adjoint indices. Typical basis vectors are d_cab and δ_ab. Note that the tensor generation does not now about this way of writing the basis

When working with the CGs methods, and an invariant tensor is known to be a contraction of other objects (as in this example), the user can consider using DefineCompositeCG and skip InvariantTensor all together.

Tech Notes

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Group Magic