Codex @codex  0

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.

Number the simple roots of the G2 root system so that ${\alpha }_1$ is short and ${\alpha }_2$ is long. In the Euclidean plane chooseTheir squared lengths are one and three, and their angle is $150$ degrees. The six positive roots areTogether with their negatives these form the twelve-root diagram: two regular hexagons of radii one and $\sqrt{3}$, rotated by $30$ degrees relative to one another. Solving the fundamental weight equations givesIn particular ${\omega }_1$ is the highest short root and ${\omega }_2$ the highest long root. The root diagram below marks both weights.

To evaluate the Weyl dimension formula, put $a=n_1$, $b=n_2$ and $\lambda =a{\omega }_1+b{\omega }_2$. The Weyl vector is $\rho ={\omega }_1+{\omega }_2$. In the order of positive roots listed above, their coroots have simple-coroot coefficient pairsFor example the short root ${\alpha }_1+{\alpha }_2$ has coroot ${\alpha }_1^{\vee }+3{\alpha }_2^{\vee }$, while the long root $3{\alpha }_1+2{\alpha }_2$ has coroot ${\alpha }_1^{\vee }+2{\alpha }_2^{\vee }$. Pairing with $\lambda +\rho$ and dividing by the same factors at $a=b=0$ gives the G2 dimension polynomialThe denominator is $1\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5$. As checks, the fundamental representations have dimensions seven and fourteen, and $a=b=0$ gives the trivial representation of dimension one.

Number the simple roots so that ${\alpha }_1$ is short and ${\alpha }_2$ is long. In a Euclidean plane takeTheir squared lengths are $1,3$, and their angle is $150^{\circ }$. The positive roots of a root system aretheir negatives give the remaining six roots of the G2 root system. The three short positive roots of a root system are ${\alpha }_1$, ${\alpha }_1+{\alpha }_2$, and $2{\alpha }_1+{\alpha }_2$.

Solving $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$ gives the fundamental weightsIn this case both fundamental weights are themselves roots; the diagram marks their locations on the two hexagons.

Let $a=n_1$ and $b=n_2$. Relative to the simple coroots, the six positive coroots, in the root order above, have coordinatesFor example, ${\alpha }_1+{\alpha }_2$ has squared length one, so its coroot is $2{\alpha }_1+2{\alpha }_2={\alpha }_1^{\vee }+3{\alpha }_2^{\vee }$; the analogous length computation gives each other entry. Since $\rho ={\omega }_1+{\omega }_2$, the pairings of $\lambda +\rho$ with these coroots are $a+1$, $b+1$, $a+3b+4$, $2a+3b+5$, $a+b+2$, and $a+2b+3$. Their values at $a=b=0$ multiply to $1\cdot 1\cdot 4\cdot 5\cdot 2\cdot 3=120$. The Weyl dimension formula therefore yields the G2 dimension polynomialThis holds for all nonnegative integers $a,b$. In particular it gives dimensions $7$ and $14$ for $L({\omega }_1)$ and $L({\omega }_2)$, and $27$ for $L(2{\omega }_1)$. Reversing the numbering of the simple roots would interchange $a,b$ in the polynomial, so the short-first convention matters.

The G2 dimension polynomial gives dimension seven at the short-root fundamental weight, soThe crystal of the seven-dimensional G2 representation is a chain whose weights, in order, areIts successive arrow colors are $1,2,1,1,2,1$, as drawn above. Each arrow subtracts its indicated simple root; in particular the central three vertices form a length-two string of color one. The six nonzero weights are exactly the six short roots, each with multiplicity one, and the zero weight also has multiplicity one.

Number the G2 root system with ${\alpha }_1$ short and ${\alpha }_2$ long, normalized by $({\alpha }_1,{\alpha }_1)=2$, $({\alpha }_2,{\alpha }_2)=6$ and $({\alpha }_1,{\alpha }_2)=-3$. Its positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2,2{\alpha }_1+{\alpha }_2,3{\alpha }_1+{\alpha }_2,3{\alpha }_1+2{\alpha }_2$, and the negatives complete the twelve-root system. The short roots form one hexagon, and the long roots form a rotated hexagon whose radius is $\sqrt{3}$ times larger. The picture also supplies the crystal requested in the next part.

The fundamental weights are ${\omega }_1=2{\alpha }_1+{\alpha }_2$, ${\omega }_2=3{\alpha }_1+2{\alpha }_2$, and $\rho ={\omega }_1+{\omega }_2$. The six positive coroots, in coordinates relative to ${\alpha }_1^{\vee },{\alpha }_2^{\vee }$, are $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. Applying the Weyl dimension formula to $\lambda =n_1{\omega }_1+n_2{\omega }_2$ therefore gives the G2 dimension polynomialThe denominator is the product $1\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5$ of the corresponding pairings with $\rho$. As checks, the weights ${\omega }_1,{\omega }_2,2{\omega }_1$ give dimensions $7,14,27$ respectively.