7 SL₃ 7.1 The Lie algebra sl_3,C 7.3 Visualising weights

7.2 Weights

Suppose that $(\rho,V)$ is a finite-dimensional complex-linear representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Suppose that $v\in V$ is a simultaneous eigenvector for all $\rho(H)$ , $H\in\mathfrak{h}$ . Then, for each $H\in\mathfrak{h}$ , there is an $\alpha(H)\in\mathbb{C}$ such that $\rho(H)v=\alpha(H)v$ . Since $\rho$ is complex-linear, $\alpha$ is a complex-linear map $\mathfrak{h}\rightarrow\mathbb{C}$ . In other words, $\alpha$ is an element of the dual space $\mathfrak{h}^{*}$ . This motivates the following definition:

Definition 7.3.

Suppose that $(\rho,V)$ is a representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Then a weight vector in $V$ is $v\in V$ such that there is $\alpha\in\mathfrak{h}^{*}$ (the weight) with:

\rho(H)v=\alpha(H)v

for all $H\in\mathfrak{h}$ .

The weight space of $\alpha$ is

V_{\alpha}=\{v\in V:\rho(H)v=\alpha(H)v\text{ for all $H\in\mathfrak{h}$}\}.

On $\mathfrak{h}$ we have some ‘obvious’ functionals $L_{i}\in\mathfrak{h}^{*}$ given by

L_{i}\left(\begin{smallmatrix}a_{1}&&\\ &a_{2}&\\ &&a_{3}\end{smallmatrix}\right)=a_{i}.

These span $\mathfrak{h}^{*}$ , subject to the relation⁶⁶ 6 More precisely, $\mathfrak{h}^{*}$ is isomorphic to the quotient of the three dimensional vector space with basis $\{L_{1},L_{2},L_{3}\}$ by the subspace spanned by $L_{1}+L_{2}+L_{3}$ .

L_{1}+L_{2}+L_{3}=0.

We compute the weights of some particular representations.

Example 7.4.

If $V=\mathbb{C}^{3}$ is the standard representation of $\mathfrak{sl}_{3,\mathbb{C}}$ (for which $\rho(A)=A$ for all $A\in\mathfrak{sl}_{3,\mathbb{C}}$ ), then the standard basis vectors $e_{1},e_{2},e_{3}$ are all weight vectors:

\begin{pmatrix}a_{1}&&\\ &a_{2}&\\ &&a_{3}\end{pmatrix}e_{i}=a_{i}e_{i}

from which we see that $He_{i}=L_{i}(H)e_{i}$ for all $H\in\mathfrak{h}$ . See table 5.

Weight	$L_{1}$	$L_{2}$	$L_{3}$
Weight vector	$e_{1}$	$e_{2}$	$e_{3}$

Table 5: Weights of the standard representation

Example 7.5.

If $V=(\mathbb{C}^{3})^{*}$ is the dual of the standard representation then it has a basis $e_{1}^{*},e_{2}^{*},e_{3}^{*}$ defined by

e_{i}^{*}(e_{j})=\delta_{ij}.

One can show that $e_{i}^{*}$ is a weight vector of weight $-L_{i}$ , so the weights are $-L_{1},-L_{2},-L_{3}$ . See problem 87.

Example 7.6.

Let $V=\operatorname{Sym}^{2}(\mathbb{C}^{3})$ be the symmetric square of the standard representation. The rules for calculating the weights of $V$ are the same as for $\mathfrak{sl}_{2,\mathbb{C}}$ — so, for the symmetric square, we add all unordered pairs of weights of $\mathbb{C}^{3}$ . For details see section 7.5 The weights of $\mathbb{C}^{3}$ are $\{L_{1},L_{2},L_{3}\}$ and so the weights of $\operatorname{Sym}^{2}(\mathbb{C}^{3})$ are

\{2L_{1},2L_{2},2L_{3},L_{1}+L_{2},L_{2}+L_{3},L_{1}+L_{3}\}.

Note that, if we wanted, we could also write $L_{i}+L_{2}=-L_{3}$ etc.

Example 7.7.

Let $V=\mathfrak{g}$ with the adjoint representation defined by $\rho(X)Y=[X,Y]$ . As already observed, we have

[H,E_{ij}]=L_{i}-L_{j}

for $H\in\mathfrak{h}$ and $i\neq j$ , while $[H,H^{\prime}]=0$ for $H,H^{\prime}\in\mathfrak{h}$ . Thus the weights of the adjoint representation are $L_{i}-L_{j}$ ( $i\neq j$ ) and $0$ . The weight space for $0$ is $\mathfrak{h}$ , which has dimension two with basis $H_{12}$ and $H_{23}$ ; we say that the weight $0$ has multiplicity two in $V$ . We obtain table 6.

\begin{array}[h]{c|ccc}\text{Weight}&\text{Weight space basis}&&\\ \@cline{1-2}\cr\ldelim.{2}{0mm}\ldelim.{3}{0mm}L_{1}-L_{2}&E_{12}&\rdelim\}{3}% {*}\hbox{\multirowsetup\text{Positive roots}}&\rdelim\}{2}{*}\hbox{% \multirowsetup Simple roots}\\ L_{2}-L_{3}&E_{23}\\ L_{1}-L_{3}&E_{13}&\\ \@cline{1-2}\cr\ldelim.{3}{0mm}L_{2}-L_{1}&E_{21}&\rdelim\}{3}{*}\hbox{% \multirowsetup\text{Negative roots}}&\\ L_{3}-L_{2}&E_{32}&\\ L_{3}-L_{1}&E_{31}&\\ \@cline{1-2}\cr 0&H_{12},H_{13}&&\end{array}

Table 6: Weights of the adjoint representation

Definition 7.8.

A root of $\mathfrak{g}=\mathfrak{sl}_{3,\mathbb{C}}$ is a nonzero weight of the adjoint representation. A root vector is a weight vector of a root, and a root space is the weight space of a root.

In other words, a root $\alpha$ with root vector $0\neq E\in\mathfrak{g}$ is a nonzero element $\alpha\in\mathfrak{h}^{*}$ such that

[H,E]=\alpha(H)E.

We write

\Phi=\{\pm(L_{1}-L_{2}),\pm(L_{2}-L_{3}),\pm(L_{1}-L_{3})\}

for the set of roots of $\mathfrak{sl}_{3,\mathbb{C}}$ . Out of these, we call $\Phi^{+}=\{L_{1}-L_{2},L_{2}-L_{3},L_{1}-L_{3}\}$ the positive roots and $\Phi^{-}=\{L_{2}-L_{1},L_{3}-L_{2},L_{3}-L_{1}\}$ the negative roots. We write $\Delta=\{L_{1}-L_{2},L_{2}-L_{3}\}$ ; these are the simple roots. Note that $L_{1}-L_{3}$ is the sum of the two simple roots. We will sometimes write $\alpha_{ij}$ for the root $L_{i}-L_{j}$ .

Finally, we have the root space or Cartan decomposition

\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha},

where the $\mathfrak{g}_{\alpha}$ are the root spaces, which are all one-dimensional.

Exercise 7.9.

Work through all the above theory in the case of $\mathfrak{sl}_{2}$ . What are the roots and root spaces? What is the relation between the weights (as linear functionals on $\mathfrak{h}$ ) and between the weights defined in section 6?