RepresentationDecomposition
RepresentationDecomposition[originalRepresentation,decomposition]
specifies the branching rules of the representations of the original group in the SSB to representations of the remnant gauge symmetry (stability group).
Details and Options
- RepresentationDecomposition can only be used after the symmetry breaking pattern has been set using SetSymmetryBreakingPattern. Only branching rules of representations of the original group can be set. Only representations of the stability group can be used in the decomposition.
- The expected input for the orignalRepresentation is
-
u1Group[1] the unit charge of an Abelian factor of the original symmetry group (all other charges are infered from the unit). nonAbelianRep the label of any multi-dimensional representation of the original group. - The expected input for the decomposition is
-
rep a single representation of the same size as the original one. {rep1, rep2,…} a list of representations understood as the direct sum of said representations. - The allowed format for the representations (rep1, rep2, …) in the branching rule/decomposition is
-
u1Group[charge] complex singlet with given charge under the U(1) factor u1Group of the stability group. nonAbelianRep the label of any multi-dimensional representation of the stability group. {factor1, factor2, …} product representation where each factor is assumed to be either representation labels or U(1) charges as above. Multiple representation factors of the same group is prohibited. Factors can be both charges and multi-dimensional representations. Singlet trivial, singlet representation. It is assumed to be real/complex according to the reality of originalRepresentation. - If the originalRepresentation is real any complex representations of the branching rules are expected to come in conjugate pairs. The two representations of such a pair must be entered immediately next to each other in the list of representations in the decomposition. This is needed for other methods to be able to pair the two, and determine the two-index invariants in the new basis.
Examples
open allclose allBasic Examples (2)
Take the EW symmetry breaking is initiated with SetSymmetryBreakingPattern:
The adjoint (real triplet) representation of 
decomposes into a complex representation with charge one and its conjugate (they follow each other in the decomposition below as they must) in addition to a real Singlet for a total of three dimensions.
This is as we expect it from the triplet gauge field 
decomposing into 
and Zμ.
The unit hypercharge becomes a unit of EM charge after EWSB, indicated by
This follows from the usual identification of EM charge as 
.
Scope (1)
In the BSM symmetry breaking scenario 
(see e.g. https://arxiv.org/abs/1808.00942)
the branching rules of the adjoint (real 15) of 
decomposes as
The complex color triplets of the decomposition are also charged under the remnant hypercharge group. The product representations are indicated with sub lists. Clearly, the charged triplets are complex, so they form a conjugate pair and must immediately follow each other in the branching rules.