7 SL₃ 7.6 Highest weights 7.8 Irreducible representations of sl_3,C.

7.7 Weyl symmetry — not examinable

Let $s_{1}$ , $s_{2}$ , and $s_{3}$ be, respectively, reflections in the lines through $L_{1}$ , $L_{2}$ , and $L_{3}$ . Then any two of these (say $s_{1}$ and $s_{3}$ ) generate the Weyl group $W$ , which is the group of symmetries of the triangle with vertices $L_{1},L_{2},L_{3}$ . So we have $W\cong D_{3}\cong S_{3}$ . Note that $W$ acts on the plane in a way that preserves the weight lattice. See Figure 10.

Theorem 7.25.

Let $(\rho,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Then the weights of $V$ are symmetric with respect to the action of the Weyl group.

Proof.

We will prove they are symmetric with respect to $s_{3}$ by using the inclusion

\iota=\iota_{12}:\mathfrak{sl}_{2,\mathbb{C}}\rightarrow\mathfrak{sl}_{3,% \mathbb{C}}

that puts a $2\times 2$ matrix in the top left corner of a $3\times 3$ matrix. We consider the restriction of $V$ to $\mathfrak{sl}_{2,\mathbb{C}}$ .

Note that if $v\in V$ is a weight vector with weight $aL_{1}-bL_{3}$ , then

\rho(\iota(H))v=\rho\left(\begin{pmatrix}1&&\\ &-1&\\ &&0\end{pmatrix}\right)v=av.

Thus $v$ is an $\mathfrak{sl}_{2,\mathbb{C}}$ -weight vector with weight $a$ . Note that $\iota(X)=E_{12}$ , so an $\mathfrak{sl}_{2,\mathbb{C}}$ -weight vector in $V$ is an $\mathfrak{sl}_{2,\mathbb{C}}$ -highest weight vector if it is killed by $E_{12}$ . Note also that $\iota(Y)=E_{21}$ .

The kernel of $E_{12}$ on $V$ is preserved by $\mathfrak{H}$ (check it!) and so has a basis made up of $\mathfrak{sl}_{3,\mathbb{C}}$ -weight vectors $v_{1},\ldots,v_{r}$ . These are then a maximal set of linearly independent highest weight vectors for $\mathfrak{sl}_{2,\mathbb{C}}$ and in particular, if $V_{i}$ is the $\mathfrak{sl}_{2,\mathbb{C}}$ -representation generated by $v_{i}$ then, as an $\mathfrak{sl}_{2,\mathbb{C}}$ -representation,

V=\bigoplus_{i=1}^{r}V_{i}.

Fix $i$ ; it suffices to show that $V_{i}$ has a basis of $\mathfrak{sl}_{3,\mathbb{C}}$ -weight vectors whose weights are preserved by $s_{3}$ . Let $v_{i}$ have weight $aL_{1}-bL_{3}$ . It follows from the $\mathfrak{sl}_{2,\mathbb{C}}$ -theory that $a\geq 0$ and — remembering that $\iota(Y)=E_{21}$ — that $V_{i}$ has a basis

v,E_{21}v,\ldots,E_{21}^{a}v.

By the FWC (Lemma 7.18) we see that these are weight vectors with weights

aL_{1}-bL_{3},(a-1)L_{1}+L_{2}-bL_{3},\ldots,L_{1}+(a-1)L_{2}-bL_{3},aL_{2}-bL% _{3}

which are symmetrical under $s_{3}$ (this reflection swaps $L_{1}$ and $L_{2}$ ), as required. This argument is illustrated in Figure 11

Invariance with respect to the other reflections is proved similarly using the other inclusions $\iota_{ij}$ of $\mathfrak{sl}_{2,\mathbb{C}}$ in $\mathfrak{sl}_{3,\mathbb{C}}$ . ∎

Corollary 7.26.

Every highest weight is of the form $aL_{1}-bL_{3}$ for $a,b\geq 0$ integers. The region

\{aL_{1}-bL_{3}:a,b\in\mathbb{R}_{\geq 0}\}

is called the dominant Weyl chamber and weights inside it (including the boundary) are dominant weights.

Proof.

Indeed, in the course of the proof of Theorem 7.25 we showed that if $aL_{1}-bL_{3}$ was a highest weight, then $a\geq 0$ . A similar argument shows that $b\geq 0$ . ∎

Exercise 7.27.

We can give another proof of Weyl symmetry using the Lie group $\operatorname{SL}_{3}(\mathbb{C})$ . Let $(\rho,V)$ is a representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . As $\operatorname{SL}_{3}(\mathbb{C})$ is simply-connected $\rho$ exponentiates to a representation, $\tilde{\rho}$ , of $\operatorname{SL}_{3}(\mathbb{C})$ . Let

\sigma_{3}=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&1\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}).

Show that, for every weight $\alpha$ , $\tilde{\rho}(\sigma_{3})$ is an isomorphism

V_{\alpha}\rightarrow V_{s_{3}\alpha}.

Give another proof of Theorem 7.25.