Crystal of the seven-dimensional G2 representation

The seven-dimensional irreducible representation of the G2 root system has highest weight ${\omega }_1=2{\alpha }_1+{\alpha }_2$. Its crystal basis is the chain of weights $2{\alpha }_1+{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1,0,-{\alpha }_1,-{\alpha }_1-{\alpha }_2,-2{\alpha }_1-{\alpha }_2$, with successive edge colors $1,2,1,1,2,1$. Each step subtracts its color's simple root. The nonzero weights are the six short roots, each of multiplicity one; the zero weight also has multiplicity one.

For $V=L({\omega }_1)$ with the G2 root system numbered short-first,In the chain basis of the crystal of the seven-dimensional G2 representation, normalize $v_1=f_1{v}_0$, $v_2=f_2{v}_1$, $v_3=f_1{v}_2$. Then $v_0\wedge v_1$ and $v_0\wedge v_3-2v_1\wedge v_2$ are nonzero highest-weight vectors of weights ${\omega }_2$ and ${\omega }_1$; the identities $e_1{v}_1=v_0$, $e_1{v}_3=2v_2$, $e_2{v}_2=v_1$ verify both raising conditions. The Weyl complete reducibility theorem and G2 dimension polynomial exhaust the twenty-one dimensions of the exterior square. In the symmetric square, $v_0^2$ generates the twenty-seven-dimensional summand. Self-duality supplies a Lie-invariant bilinear form, which is symmetric because a nondegenerate alternating bilinear form cannot have odd dimension; its inverse gives the remaining invariant line.