Re: Sphere packing in six dimensions, root system for E6, Wikipedia article


On 7-Oct-2009, David Bernier <david250@xxxxxxxxxxxx>
wrote in message <haht7q02e85@xxxxxxxxxxxxxxxxx>:

Jim Heckman wrote:

On 4-Oct-2009, David Bernier <david250@xxxxxxxxxxxx>
wrote in message <haaud101ik9@xxxxxxxxxxxxxxxxx>:

[...]

Then I believe I tried all pairs
{alpha, beta} in Phi_E8 with alpha and beta orthogonal,
counting orthogonal roots as usual. That didn't seem to work,
but I wasn't sure why.

By the way, I forgot to mention that choosing alpha and beta
orthogonal should always give 60 roots orthogonal to both, in the
D6 configuration. This is because what you're doing is equivalent
to finding the stabilizer of a root in E7. Specifically, the
stabilizer of alpha is an E7 which contains beta, and then you're
looking for the stabilizer of beta within that E7.

Then I tried {alpha, beta} with
2 < alpha, beta >/<alpha, alpha> = -1 (pi/3 angle), and that
gave 72 vectors, then I gave an answer to the kissing balls
question. [ But after reading your first reply,
I thought I might have made mistakes somewhere, or
forgotten some detail.]

[...]

Thanks for your explanations. I think the article on root systems
doesn't say much about their connection with Lie Theory.

There's a book by Armand Borel,
"Essays in the history of Lie groups and algebraic groups",
that was published in 2001.

Thanks, I may take a look at that. Actually, my knowledge of Lie
algebras and groups per se is pretty meager. But I do know somewhat
more about the finite Coxeter groups, which include the Weyl groups
of the semisimple Lie algebras (plus some others that aren't Weyl
groups because they don't satisfy the crystallographic condition).

[...]

--
Jim Heckman
.

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