I have recently irritated a number of long-time contributors to this site by:
i) posting multiple (but not duplicate) questions involving the roots of the algebraic-group $E_8$
2) posting this relatively large number of questions over a relatively short period of time.
3) posting these questions without providing sufficient context that would explain my reason for posting them.
So, I want to first apologize for (1-3).
But I also want to very sincerely say "thank-you" to this site for allowing me to post these questions, for the following reason.
The reason I posted these questions was to try and get insight into a very specific problem confronting the research team of which I am a member. This problem was very simple, and can be stated as follows:
4) My team has a set of 240 elements which can be split into two groups of (144,96) in a principled way, i.e. this split is "principled" because it arises as a natural and inexorable consequence of certain fundamental biomolecular "first principles" on which all molecular biologists agree
5) My team also has several reasons to believe that these 240 elements do, in fact, instantiate the roots of $E_8$.
6) but our problem arose because we could not see how our intuitively obvious and principled split of the 240 into (144,96) could be reconciled with the usual split of the 240 into (128,112), i.e. the split according to the usual coordinatization of the roots of $E_8$ as given in the "Construction" section here:
And all of the "odd" questions which I recently posted about the roots of $E_8$ were in fact posted in an attempt to gain some insight into how this reconciliation could be accomplished.
And this attempt succeeded, but NOT for the reason one might expect, i.e. I did not gain insight as a result of any answers to the questions I posted.
Instead, the very process of posting these questions forced me to sharpen my thinking about our (144,96) split, and as a result, I was able yesterday to finally see that there is a natural and principled way to reconcile this split with the usual (128,112) split.
And this is why I want to express my deepest gratitude to this site. If this site did not exist as a place where I could sharpen my thinking about our (144,96) split by posting questions about $E_8$ to multiple knowledgeable communities, I truly believe that the reconciliation of our (144,96) split with the standard (128,112) split would have taken a lot longer. And in fact it might well be that I never would have seen how to accomplish this reconciliation.
So, thank you so much for tolerating my presence here over the past few weeks.
I owe this site an intellectual debt which I will never be able to repay.
And if you want to know why I feel that way, it is because my team can now:
i) CERTAINLY show that biomolecular codon space (the space of genes) instantiates two opposed instances of 4$_2$$_1$, while biomolecular amino acid space (the space of proteins encoded by genes) instantiates two opposed instances of 1$_2$$_2$
ii) PROBABLY show that biomolecular codon space instantiates two opposed copies of $E_8$ (AS WELL AS their associated Coxeter groups), while biomolecular amino acid space also instantiates two opposed copies of $E_6$ (AS WELL AS their associated Coxeter groups).
And since we can now make those two claims with some degree of confidence that they might be correct, we can also make a third claim which goes to the "heart of the matter"
iii) the particular way in which amino acids are encoded by codons (and proteins by genes) can PROBABLY be shown to follow as a natural consequence of the particular way in which the 72 roots of $E_6$ occur as a subset of the 240 roots of $E_8$.