Trace of $SU(N)$ generators and number of spin components

Question

I am reading Peskin's and Schroeder's book "An Introduction to Quantum Field Theory". At Chapter 16.6, the authors write down an expression for the trace of of two generators of $SU(N)$ $$\text{tr}[t^at^b]=C(r)d(j)\delta^{ab}\tag{16.124}$$ where $C(r)$ is a constant that depends on the representation and $d(j)$ is "the number of spin components". This is different than the expression in Chapter 15.4, $$\text{tr}[t_r^at_r^b]=C(r)\delta^{ab}\tag{15.78}$$ where now $t_r$ stands for the generator of $SU(N)$ in the irreducible representation $r$.

My question is the following: I can see that the factor $d(j)$ is the only difference between working in a reducible (I presume that in the first case the representation is reducible) representation and an irreducible one. Why is it different for reducible representations and why the difference is simply a factor equal to the number of spin components? Can someone give a concrete example of such representations say for $SU(2)$?

Answer 1

P&S are in eq. (16.122) essentially calculating the one-loop contribution to the gluon self-energy/vacuum polarization in QCD, where we should sum over all possible fields running in a loop. This produces a trace over both spin indices and internal gauge indices for the field in the loop. Eq. (16.124) merely reflects this.