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I need to show that the following are subgroups of $E_8$ using extended Dynkin diagrams.

$$SU\left(5\right)\times SU\left(5\right)$$ $$SU\left(3\right)\times E_6$$ $$SU\left(4\right)\times SO\left(10\right)$$ and $$SO\left(16\right)$$ $$SU\left(2\right)\times E_7$$ $$SU\left(9\right)$$

Is it enough to find the Dynkin diagrams for the subgroups by deleting edges in the Dynkin diagram for $E_8$? If so, I can only seem to do this for the first ones listed.

Is the use of extended Dynkin diagrams important?

Joshua Tilley
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1 Answers1

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You have to remove vertices, and then all edges which connect to such a removed vertex.

You must have done that wrongly, because you get none of the groups you list by removing something from the original Dynkin diagram. Notice that they all have rank $8$, whereas if you remove a vertex from a diagram of type $E_8$, you're left with something of rank $\le 7$.

From the extended diagram however (which looks like something one might call "$E_9$"), one can remove one of the vertices and its connecting edges to get diagrams of type $A_4 \times A_4, A_2 \times E_6, A_3 \times D_5, D_8, A_1 \times E_7$ and $A_8$, respectively, which correspond to the six groups you list.

So no, it is not enough to use the non-extended diagram, and yes, it is important.

Torsten Schoeneberg
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  • Ahh, I see, thanks! I was removing an edge (not removing any vertices) from the Dynkin diagram (not extended). I'm now not clear what exactly constitutes an extended Dynkin diagram, it appears to be one with an extra vertex added to the usual diagram and that vertex is connected to some of the original diagram's vertices. Does it have to attach to particular vertices/ with a particular number of connections? – Joshua Tilley Nov 27 '18 at 21:19
  • Yes. The extra vertex in the extended diagram corresponds to the negative of the highest root w.r.t. the basis given by the diagram. Every good book on the matter should contain a table of the extended diagrams. Wikipedia has one: https://en.wikipedia.org/wiki/Dynkin_diagram#/media/File:Affine_Dynkin_diagrams.png Note that in type $E_6$, the extra vertex docks at a different place than in $E_7$ and $E_8$. – Torsten Schoeneberg Nov 28 '18 at 01:15