I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a little less vague, I should indicate what I mean by "real world": First of all, in physics, I require that this be experimentally verifiable, so $G_2$ holonomies in stringy math should not count, nor should grand unified $E8$-theories. Industrial applications would be great, as would examples from biology.
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asked Sep 9, 2023 at 8:51
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8$\begingroup$ Do uses of the E8 lattice in error-resistant communications count? $\endgroup$– Noam D. ElkiesCommented Sep 9, 2023 at 11:34
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5$\begingroup$ Baez and Huerta - $G_2$ and the rolling ball. $\endgroup$– LSpiceCommented Sep 9, 2023 at 13:33
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7$\begingroup$ I have no idea what you mean by the “real world”, but whatever it is math is part of it. $\endgroup$– Andy PutmanCommented Sep 9, 2023 at 14:38
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5$\begingroup$ @AndyPutman: I tried to define "real world" in the question. $\endgroup$– Lorenzo Del VecchiopontopolosCommented Sep 10, 2023 at 10:29
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3$\begingroup$ @AndyPutman How many times have I heard a mathematician say "I have no idea what you mean by X," or slight variations of it, where X is just a perfectly plain concept understandable by everyone. Typically they are pretty smug about it, too. But if you really want a fruitful conversation with someone, you should at least attempt to construct a generous interpretation of what the other person says. $\endgroup$– R.P.Commented Sep 17, 2023 at 16:35
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1 Answer
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As Noam Elkies alluded to in a comment, the E8 lattice plays a role in coding theory, basically because it such an efficient sphere packing. For example, Kurkoski has proposed using it for error correction in flash memory: The E8 Lattice and Error Correction in Multi-Level Flash Memory (but I don't know if this error-correction scheme is actually used in practice).
answered Sep 10, 2023 at 19:20