Group representation definition

Question

I'm struggling to understand a passage of definition in group representation on Pierre Ramond's book "Group Theory: A physicist's surveys".

My doubt is in this definition of (3.6). I can understand that 3.5 is about homomorphism, but about 3.6 I could not figure out.

Thank you for your attention.

Answer 1

Formulas (3.5) and (3.6) together rephrase the fact that $$ \begin{pmatrix} \mathcal{M}^{[1]}(gg') & 0 \\ \mathcal{N}(gg') & \mathcal{M}^{[\perp]}(gg') \end{pmatrix} = \begin{pmatrix} \mathcal{M}^{[1]}(g) & 0 \\ \mathcal{N}(g) & \mathcal{M}^{[\perp]}(g) \end{pmatrix} \cdot \begin{pmatrix} \mathcal{M}^{[1]}(g') & 0 \\ \mathcal{N}(g') & \mathcal{M}^{[\perp]}(g') \end{pmatrix}.$$