Talk:F4 (mathematics)

F4 lattice?

Is this the same as the icositetrachoronic tetracomb with Schläfli symbol {3,4,3,3}? Tom Ruen 17:01, 5 June 2007 (UTC)[reply]

No, the vertices of {3,4,3,3} do not form a lattice. Rather, the vertices of the dual {3,3,4,3} form an F_4 lattice. The situation is analogous to the two-dimensional cases {6,3} and {3,6}. -- Fropuff 16:45, 6 June 2007 (UTC)[reply]

Thanks! Good to know. Is it worth linking to hexadecachoronic tetracomb {3,3,4,3}? Or equally I wonder - should F4 lattice instead be redirect to this honeycomb? Tom Ruen 18:23, 6 June 2007 (UTC)[reply]

And the third one is the second one

"There are 3 real forms: a compact one, a split one, and a third one."

Sorry but to me that is a very funny sentence! Moon Oracle (talk) 18:28, 10 April 2012 (UTC)[reply]

Maybe I was the one who wrote this. If so, I guess I couldn't find a better way to phrase it: there are three real forms of F₄, one of them is remarkable in that it is the compact form, another is remarkable in that it is the split form, and the third form doesn't have such a convenient one-word description. Feel free to rewrite more eloquently. ;-) (Of course, we should really be writing a full section on real forms and describe them in some detail.) --Gro-Tsen (talk) 21:44, 10 April 2012 (UTC)[reply]

Invariant polynomials

I deleted the two sentences from current version of the page (28 October 2014): "F_4 is the only exceptional lie group which gives the automorphisms of a set of real commutative polynomials. (The other exceptional lie groups require anti-commutative polynomial invariants)." This is not true: for example, the 3875-dimensional irreducible representation of E₈ has an invariant quadratic form and an invariant cubic polynomial, and the subgroup of GL₃₈₇₅ stabilizing these two polynomials is exactly E₈. Exceptg (talk) 18:38, 28 October 2014 (UTC)[reply]

Ah, I always found this statement suspicious, but I wasn't sure enough to remove it. I'm glad to know my suspicion was founded. But is something like what was claimed perhaps true for the lowest-dimensional faithful representation? Also, does your statement about the 3875-dimensional representation of E₈ follow from an explicit computation, or is it a standard fact, or what? If you have references about this sort of things, maybe you could add them to the article (and its relatives about the other exceptional Lie groups). --Gro-Tsen (talk) 11:13, 10 November 2014 (UTC)[reply]

Well if it is true then it is very new stuff. I found this paper http://arxiv.org/pdf/1309.6611.pdf which is dated 28 April 2015 (two weeks ago!) — Preceding unsigned comment added by 94.119.106.25 (talk) 17:17, 12 May 2015 (UTC)[reply]