Group representations
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- Clemence Mosley
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1 Group representations A 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) structure constants (real numbers) based on S-70 the dimension of the representation taking the complex conjugate we see that R is real if or if there is a unitary transformation is also a representation! e.g. fundamental reps. of SO(N) that makes R is pseudoreal if it is not real but there is a transformation such that if R is not real or pseudoreal then it is complex complex conjugate representation is specified by: e.g. the fundamental rep of SU(2) : e.g. fundamental reps. of SU(N), N>2, 243
2 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
3 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
4 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
5 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
6 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
7 generator matrices for are then given by and an infinitesimal group transformation of we trade complex conjugation for transposition is: is invariant! 249
8 Consider the Kronecker delta symbol is an invariant symbol of the group! this means that the product of the representations and must contain the singlet representation, specified by. Thus we can write: 250
9 Another invariant symbol: this implies that: must contain the singlet representation! 251
10 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
11 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
12 Consider now a pseudoreal representation: R is pseudoreal if it is not real but there is a transformation such that 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
13 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.: normalized so that for SU(N) with. is completely symmetric Since we have: we also have: for real or pseudoreal representations. e.g. for SU(2), all representation are real or pseudoreal and for all of them 255
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