# Are the 240 roots of $E_8$ in ANY special relationship with any PARTICULAR set(s) of 1296 elements of $E_8$?

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Are the 240 roots of $E_8$ in ANY special relationship with any PARTICULAR set(s) of 1296 elements of $E_8$?

If the answer to this question is "yes", then I can probably go away and stop bothering you good people, at least for a few weeks.

Because if the answer is "yes", then the answer will probably tell me exactly how $E_8$ is instantiated in the energetics of the particular biomolecular subsystem which my team has been studying for quite a number of years.

So, if you'd like me to go away, at least for a while, please take a moment to think about this question. And thanks as always for whatever time you can afford to spend doing so.

• I don't see anything right away. The next shell on the $E_8$-lattice has 2160 vectors. I just counted them myself to make sure they didn't miss any! I don't see this number in the theta series of E6 either. I'm afraid I don't really know the polytope stuff. I was rather expecting that you describe the set of vectors (or molecules even!) that you think have some large Weyl group (or a subgroup) as its group of symmetries. Dec 3, 2017 at 22:23
• @JyrkiLahtonen - thank you so much!!!! - believe it or not, I know exactly what your 2160 "means" in terms of the model we're developing!!!! But I wouldn't have known this until last night because it wasn't until last night that I realized how to reconcile a critical (144,96) split of 240 in our model with the standard split of (128,112) given by the usual coordinatization of $E_8$. And the number 2160 is implied by the nature of that reconciliation in several different ways. I will post back on this when I have more details , but I think you've put me on the right track - THANK YOU!!!!!! Dec 3, 2017 at 22:45