It is well known the existence of a T duality between the two heterotic string theories, $SO(32) \sim\_T E\_8 \times E\_8$. Beyond the trivial point that both groups have the same dimension (496, which actually is a prerequisite), is there some other mathematical relation between them?


I am thinking in other SO(N) groups whose dimension is a perfect number and that happen to be related to products of manifolds. $SO(4)$ with $SU(2) \times SU(2)$, and -I am told- $SO(8)$ with some variant of $(S^7 \times S^7) \times G\_2$. It should be nice if all of these were justified by a common construction, but I am happy just with an answer to the $SO(32)$ case.



Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact?
The answer to this question can be found in Lubos Motl's answer to [this question of mine on Physics.SE](https://physics.stackexchange.com/q/65092/23119). 


The key here are the weight lattices bosonic representations $\Gamma$ of these gauge groups.


As I understand it, the weight lattice of $E(8)$ is $\Gamma^8$, whereas the weight lattice of $\frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}\_2}$^ is $\Gamma^{16}$. The first fact means that the weight lattice of $E(8)\times E(8)$ is $\Gamma^{8}\oplus\Gamma^8$, 


Now, an identity, that $\Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}=\Gamma^{16}\oplus\Gamma^{1,1} $ , which actually allows this T-Duality. Now, this means that it is *this very identity* which allows the identity mentioned in the original post. 


So, the answer to your question is "**Yes**", there *is* a group-theoretical fact, and that is that $ \Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16}\oplus\Gamma^{1,1} $. 


