A friend of mine shows me a decomposition,
(1). For SO(10), we have $$ 16^+ \to (2^+,4^+) \oplus (2^-,4^-) $$
(2). For SO(18), we have $$ 256^+ \to (16^+,8^+) \oplus (16^-,8^-) . $$
- I wonder what does this decomposition precisely mean?
My trial:
For (1), I think that $16^+$ is the spinor representation of SO(10), where one has $32=16^+ \oplus 16^-$.
- But what does $16^+$ and $16^-$ mean precisely?
I think the 2 has something to do with SO(4), while the 4 has something to do with SO(6).
- What do this 2 and 4 mean?
For (2), I think that $256^+$ is the spinor representation of SO(16), where one has $512=256^+ \oplus 256^-$.
- But what does $256^+$ and $256^-$ mean precisely? How does one project to two sectors?
I think the 16 has something to do with SO(10), while the 8 has something to do with SO(8).
- In this decomposition, why are we able to get the spinor representation of SO(10) on one hand, while getting the vector representation of SO(8) on the another hand?
Your question is so oracular that it would not be possible to answer unless one had prior recognition handles on GUTs.
Fortunately, from Tables 43 and 56, respectively, of R Slansky's standard 1981 review, you see that
Under SO(10) ⊃ SO(4) × SO(6) ~ SU(2) × SU(2) × SU(4) , you get the Weyl spinor irrep 16 decomposing to (2,1,4) ⊕ (1,2,$\bar 4$) .
Under SO(18) ⊃ SO(8) × SO(10), you get the spinor irrep 256 decomposing to (8,16) ⊕ (8,$\bar {16}$).
Basically, the irreducible spinor decompositions reduce to direct sums of the spinor irreps of the orthogonal subgroups specified here. The Dynkin labels, modalities and logic of the decompositions are detailed in the review and tables.