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Group representations based on S 70 A representation of a group is specified by a set of hermitian matrices that obey the dimension of the representation the original set of NxN dimensional matrices for SU N or SO N corresponds to the fundamental representation taking the complex conjugate we see that R is real if or if there is a unitary transformation R is pseudoreal if it is not real but there is a transformation such that structure constants real numbers is also a representation e g fundamental reps of SO N that makes e g the fundamental rep of SU 2 if R is not real or pseudoreal then it is complex e g fundamental reps of SU N N 2 complex conjugate representation is specified by 243 The adjoint representation A A is a real representation the dimension of the adjoint representation of generators the dimension of the group to see that s satisfy commutation relations we use the Jacobi identity follows from 244 The index of a representation The quadratic Casimir multiplies the identity matrix commutes with every generator homework S 69 2 Useful relation SU N SO N 245 A representation is reducible if there is a unitary transformation that brings all the generators to the same block diagonal form with at least two blocks otherwise it is irreducible For example consider a reducible representation R that can put into two blocks then R is a direct sum representation and we have 246 Consider a field that carries two group indices then the field is in the direct product representation The corresponding generator matrix is and we have to prove this we use the fact that 247 We will use the following notation for indices of a complex representation hermitian conjugation changes R to and for a field in the conjugate representation we will use the upper index we write generators as indices are contracted only if one is up and one is down an infinitesimal group transformation of is 248 generator matrices for are then given by we trade complex conjugation for transposition and an infinitesimal group transformation of is is invariant 249 Consider the Kronecker delta symbol is an invariant symbol of the group this means that the product of the representations the singlet representation specified by and must contain Thus we can write 250 Another invariant symbol this implies that must contain the singlet representation 251 multiplying by A we find A is real combining it with a previous result we get the product of a representation with its complex conjugate is always reducible into a sum that contains at least the singlet and the adjoint representations For the fundamental representation N of SU N we have no room for anything else 252 Consider a real representation implies the existence of an invariant symbol with two R indices For the fundamental representation N of SO N we have corresponds to a field with a symmetric traceless pair of fundamental indices 253 R is pseudoreal if it is not real but there is a transformation such that Consider now a pseudoreal representation still holds but the Kronecker delta is not the corresponding invariant symbol the only alternative is to have the singlet appear in the antisymmetric part of the product For SU N another invariant symbol is the Levi Civita symbol with N indices For SU 2 similarly for we can use and to raise and lower SU 2 indices if is in the representation then we can get a field in the representation by raising the index 254 Another invariant symbol of interest is generator matrices in any rep are invariant or the right hand side is obviously invariant Very important invariant symbol is the anomaly coefficient of the rep is completely symmetric normalized so that Since for SU N with we have for real or pseudoreal representations e g for SU 2 all representation are real or pseudoreal and for all of them we also have 255

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