= G2 dimension polynomial
{c}
{title2=$\dim L(a\omega_1+b\omega_2)$}
For the <G2 root system> with $\alpha_1$ short and $\alpha_2$ long, the <fundamental weights> are $\omega_1=2\alpha_1+\alpha_2$ and $\omega_2=3\alpha_1+2\alpha_2$. The positive <coroots>, relative to the <simple coroots>, have coordinates $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. The <Weyl dimension formula> therefore gives, for nonnegative <integers> $a,b$,
$$
\dim L(a\omega_1+b\omega_2)=\frac{(a+1)(b+1)(a+b+2)(a+2b+3)(a+3b+4)(2a+3b+5)}{120}.
$$
The denominator is the product of the pairings of the <Weyl vector> with those coroots. The <dimensions> at $\omega_1,\omega_2,2\omega_1$ are seven, fourteen and twenty-seven.
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