Transformation of the adjoint representation

[ansgar]

Homework Statement

given that

[tex]N\otimes\bar{N} = 1 \oplus A [/tex]

consinder the SU(2) subgroup of SU(N), that acts on the two first components of the fundamental representation N of SU(N). Under this SU(2) subgroup, the repsentation N of SU(N) transforms as [itex] 2 \oplus (N-2) [/itex]

with info above, how does the adjoint representation transform under this SU(2) subgroup?

The Attempt at a Solution

what does it mean that the representation transforms?

does it mean if I take one generator of the fundamental representation call it [tex] T^a [/tex]

that it transfors as [tex] T^a \rightarrow \sigma \, \lambda \, T^a [/tex]

where sigma is a SU(2) transformation matrix and lambda a SU(N-2) transf. matrix?

 

[Nino MMP]

if this is the case, then how can I find out how the adjoint representation transform under this SU(2) subgroup?