G2 Representation Theory: A Simple Question on Symmetry & Dual Groups

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I really don't know that much representation theory or group theory.

I was looking at Garrett lisi's presentation, and I was looking at the 3d root system of g2. It struck me that it looked similar to a dodecahedron inscribed in a cube. Now, I do know that the dodecahedron group, and the cube group are dual. Can g2 be broken up into the symmetries of a cube, and a dodecahedron?
 
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I meant dual Octahedron is dual to the cube. sorry
 
The Root System G_2

Jim Kata said:
I was looking at Garrett Lisi's presentation, and I was looking at the 3d root system of g2.

Unless I misunderstand what you are talking about, I think you mean 2d; see James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, Fig. 1 on p. 44, for a picture of the root system G_2.
 

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