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I think this is more stackexchange-worthy, but here goes. I'm a little afraid you're mixing up general weight diagrams with the root system, which is the weight diagram of the adjoint representation. …

$SO(6) = SU(4)/Z_2$ (i.e. the $Alt^2$ rep of $SU(4)$ preserves an $\mathbb R^6$ inside that $\mathbb C^6$), by the way. Your subgroup is of the same rank as the whole, so by Borel-de Siebenthal theo …

Topology of Lie Groups, I and II certainly has the homotopy groups up a ways.

I often find it more useful to say $SU(3)$ is a $T^2$ bundle over the manifold of flags in ${\mathbb C}^3$ (itself a ${\mathbb CP}^1$-bundle over ${\mathbb CP}^2$). Partly this is because $T^2$'s homo …

Let $M = {\mathbb R}^1$ be the time-line, with the trivialized bundle $M \times {\mathbb R}^1$ on it. There's a connection on it, called "inflation", such that parallel transport of $N$ at time $t$ to …

Every irrep of $SU(n)$ extends to irreps of $U(n)$, and conversely, the restriction of any irrep of $U(n)$ to $SU(n)$ remains irreducible. If your dominant weight of $SU(n)$ is $(a_1,\ldots,a_{n-1})$ …