Group theoretical links between SO(32) and $E_8 \times E_8$?

Question

T-duality is a canonical way to go from SO(32), or Spin(32) to $E_8 \times E_8$ and back. This is mentioned in some answers, e.g. T-duality between $E_8 \times E_8$ and $\text{Spin(}32)/\mathbb{Z}_2$ heterotic strings at the $\sigma$-model level

Is there some way to link both groups using only representation theory? For instance, SO(32) is decomposed to SU(16) or to SO(16)xSO(16) looks respectively as

$$496 = (255)_0 ⊕ (1)_0 ⊕ (120)_4 ⊕ (\overline {120})_{−4}$$ $$496 = (16, 16) ⊕ (120, 1) ⊕ (1, 120)$$

very much as two copies of $E_8$, and of course we have $E_8 \times E_8$ itself branching to $SO(16)$ as

$$248 ⊕ 248 = (128′)⊕ (128′) ⊕(120) ⊕(120)$$

And also, if we have built the 496 of $SO(32)$ as a symmetrized pairing of 16 + 16 "particles and antiparticles", it can be further splitted to $256 + 240$, which is a more informal statement of the above branchings, and again looks as two copies of SO(16).

This kind of coincidencies is usually mentioned as lore (say Baez' TWF and similar) but rarely more substance is given. Is there more content here, thus? Such as actually defining $E_8$ as some action in those vector spaces that are also representations of subgroups of $SO(32)$? And viceversa?

Also, is $SO(16) \times SO(16)$ the only maximal common subgroup useful for this sort of descriptions?

This post imported from StackExchange Physics at 2017-08-11 12:39 (UTC), posted by SE-user arivero

Comments

They share rank and order. What can you say about the difference of their Dynkin diagrams? Due diligence: re-read your Slansky. I very strongly suspect you'd find more takers in the math cousin of this site.

This post imported from StackExchange Physics at 2017-08-11 12:39 (UTC), posted by SE-user Cosmas Zachos