7 SL₃ 7.3 Visualising weights 7.5 Tensor constructions

7.4 Representations and weights

Firstly, we recall from above that any finite-dimensional representation of $\mathfrak{sl}_{3,\mathbb{C}}$ is completely reducible. This is Theorem 6.30 above, that we proved using the ’unitary trick’.

Theorem 7.13.

Let $(\rho,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Then:

•

There is a basis of $V$ consisting of weight vectors.
•

Every weight of $V$ is in the weight lattice $\Lambda_{W}$ .

We may combine these two into the single equality

V=\bigoplus_{\alpha\in\Lambda_{W}}V_{\alpha}.

Proof.

Consider the embedding $\iota_{12}:\mathfrak{sl}_{2,\mathbb{C}}\rightarrow\mathfrak{sl}_{3,\mathbb{C}}$ embedding a $2\times 2$ matrix into the ‘top left’ of a $3\times 3$ matrix:

\iota_{12}:\begin{pmatrix}a&b\\ c&-a\end{pmatrix}\mapsto\begin{pmatrix}a&b&0\\ c&-a&0\\ 0&0&0\end{pmatrix}.

This is a Lie algebra homomorphism, and $\rho\circ\iota_{12}$ is a representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . We know from the $\mathfrak{sl}_{2,\mathbb{C}}$ theory that

(\rho\circ\iota_{12})(H)=\rho\begin{pmatrix}1&&\\ &-1&\\ &&0\end{pmatrix}=\rho(H_{12})

is diagonalizable with integer eigenvalues. If $v\in V$ is a weight vector with weight $a_{1}L_{1}+a_{2}L_{2}+a_{3}L_{3}$ , then it is an eigenvector for $\rho(\iota_{12}(H))$ with eigenvalue $a_{1}-a_{2}$ . Thus $a_{1}-a_{2}$ is an integer.

Now, there is another embedding $\iota_{23}$ putting a $2\times 2$ matrix in the ‘bottom right’ corner. The same argument then shows that $\rho(H_{13})$ is diagonalisable with integer eigenvalues, which shows that $a_{2}-a_{3}$ is an integer for every weight.

Thus every weight is in the weight lattice. Moreover, $\rho(H_{12})$ and $\rho(H_{23})$ are diagonalizable, and they commute with each other since $H_{12}$ and $H_{23}$ commute and $\rho$ is a Lie algebra homomorphism. A theorem from linear algebra states that commuting, diagonalizable matrices are simultaneously diagonalizable. It follows that there is a basis of $V$ consisting of simultaneous eigenvectors for $\rho(H_{12})$ and $\rho(H_{23})$ . Since $H_{12}$ and $H_{23}$ span $\mathfrak{h}$ , this is a basis of weight vectors. ∎

Remark 7.14.

There is a third homomorphism

\iota_{23}:\begin{pmatrix}a&b\\ c&-a\end{pmatrix}\mapsto\begin{pmatrix}a&0&b\\ 0&0&0\\ c&0&-a\end{pmatrix}.

We have $\iota_{23}(H)=H_{23}$ .

Note that, for $i<j$ , $\iota_{ij}(X)=E_{ij}$ and $\iota_{ij}(Y)=E_{ji}$ , so $E_{ij}$ and $E_{ji}$ will play the role of raising and lowering operators.

Remark 7.15.

We could also prove Theorem 7.13 by exponentiating $\rho$ to a representation of $\operatorname{SL}_{3}(\mathbb{C})$ and considering the action of the subgroup of diagonal matrices with entries in $U(1)$ , which is compact (isomorphic to $U(1)$ ). Compare the proof of the statement that the weights of representations of $\mathfrak{sl}_{2,\mathbb{C}}$ are integers.