I'm studying the book "Quarks and Leptons: An Introductory Course In Modern Particle Physics". When I reach chapter 2 (page 33), I encounter this exercise problem that I couldn't understand:
EXERCISE 2.1 Justify the decomposition shown in (2.1) by either (1) considering the symmetry of the states under interchange of the nucleons or (2) using the angular momentum "step-down" operator.
Here's the solution in the book:
My questions are: Why do these "justify" the decomposition in (2.1)? Why the state $|S=1,M_S=0>$ is said to be "also" symmetric? What does it mean that the state $|S=0,M_S=0>$ be antisymmetric due to orthogonality? I'm really confused...
I'm really new to the subject, so it'd be great if someone could help me. Thank you!