in p. 424 of this book, it says that $$ n(2n-1) \text{ anti-sym rep of } SO(2n)$$ $$\to n^2-1 \text{ adjoint-rep of } SU( n)$$ $$\oplus 1 \text{ of } SU( n)$$ $$ \oplus n(n-1)/2 \text{ anti-sym rep of } SU( n)$$ $$\oplus n^*(n^*-1)/2 \text{ anti-sym rep of } SU( n) $$
How do we justify the calculations from eq.12?
I can follow the counntings, but not sure the part of $$ n \otimes_A n^* =? $$ $$ n^* \otimes_A n =? $$
Text from the book:
