Group representationsbased on S-70A representation of a group is specified by a set of hermitian matrices that obey:(the original set of NxN dimensional matrices for SU(N) or SO(N) corresponds to the fundamental representation)the dimension of the representationstructure constants (real numbers)taking the complex conjugate we see that is also a representation!R is real if or if there is a unitary transformation that makesR is pseudoreal if it is not real but there is a transformation such thatcomplex conjugate representation is specified by:e.g. the fundamental rep of SU(2) :e.g. fundamental reps. of SU(N), N>2e.g. fundamental reps. of SO(N)if R is not real or pseudoreal then it is complex , 2 43The adjoint representation A:A is a real representationthe dimension of the adjoint representation, = # of generators= the dimension of the groupto see that s satisfy commutation relations we use the Jacobi identity:follows from:2 44The quadratic Casimir :The index of a representation :multiplies the identity matrixcommutes with every generator, homework S-69.2Useful relation:SU(N): SO(N):2 45A representation is reducible if there is a unitary transformationthat 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:2 46Consider a field that carries two group indices :to prove this we use the fact that .then the field is in the direct product representation:The corresponding generator matrix is:and we have:2 47hermitian conjugation changes R to and for a field in the conjugate representation we will use the upper indexWe will use the following notation for indices of a complex representation:we write generators as:indices are contracted only if one is up and one is down!an infinitesimal group transformation of is:2 48generator matrices for are then given bywe trade complex conjugation for transpositionand an infinitesimal group transformation of is:is invariant!2 49this means that the product of the representations and must contain the singlet representation , specified by .Consider the Kronecker delta symbolis an invariant symbol of the group!Thus we can write:2 50Another invariant symbol:this implies that:must contain the singlet representation!2 51multiplying by A we find:(A is real)combining it with a previous result we getthe 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)2 52Consider a real representation :implies the existence of an invariant symbol with two R indicesFor the fundamental representation N of SO(N) we have:corresponds to a field with a symmetric traceless pair of fundamental indices2 53Consider now a pseudoreal representation:still holds but the Kronecker delta is not the corresponding invariant symbol:R is pseudoreal if it is not real but there is a transformation such thatthe 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:similarly for .For SU(2):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: .2 54for real or pseudoreal representations .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 symmetricSince we have:we also have:e.g. for SU(2), all representation are real or pseudoreal and for all of themnormalized so that for SU(N) with .2