Equations

By | 2017-09-10

Let \{\, e_i : 1 \leq i \leq 3 \,\} be an orthonormal basis. Let \mathfrak{A}_3 denote the group of cyclic permutations of \{\, 1,2,3 \,\}. Then
\{\, e_i - e_j : 1 \leq i \neq j \leq 3 \,\}
\{\, 2e_i - e_j - e_k : \left( i.j,k \right) \in \mathfrak{A}_3  ,\}
\{\, -2e_i + e_j + e_k : \left( i.j,k \right) \in \mathfrak{A}_3  ,\}
is the standard representation of a Root System of type G_2
with basis
B = \{\, \alpha_1 = e_1 - e_2 , \alpha_2 = e_2 + e_3 - 2e_1 \,\}
The set of simple roots orthogonal to the Highest Root is \left\{\alpha_1 \right\} and the corresponding Parabolic set is B_- \sqcup \left\{ \alpha_1 \right\}. Its complement is a 5-dimensional Heisenberg Set

3\alpha_1 + 2\alpha_2
\alpha_2 3\alpha_1 + \alpha_2
\alpha_1 + \alpha_2 2\alpha_1 + \alpha_2

The 4 maximal Abelian subsets are permuted by the canonical involution of the Heisenberg Radical hence we may choose as representatives those which contain \alpha_2:
\{\, \alpha_2 , 2\alpha_1 + \alpha_2, 3\alpha_1 + 2\alpha_2  \,\}
\{\, \alpha_2 , \alpha_1 + \alpha_2, 3\alpha_1 + 2\alpha_2  \,\}
The Abelian relations for these sets are generated by
3(2\alpha_1 + \alpha_2) - 2(3\alpha_1 + 2\alpha_2) + \alpha_2 = 0
3(\alpha_1 + \alpha_2) - (3\alpha_1 + 2\alpha_2) - \alpha_2 = 0
The corresponding polynomial relations in variables X,Y,Z are
X^3Y=Z^2
XY=Z^3

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