7 SL₃ 7.4 Representations and weights 7.6 Highest weights

7.5 Tensor constructions

We did not go through these proofs in lectures in detail, so I only expect you to know how to apply these results in situations similar to those in lectures or in the problems class.

We record how the various linear algebra constructions we know about interact with the theory of weights. If $(\rho,V)$ is a representation of $\mathfrak{g}$ then we consider its weights as a multiset

\{\alpha_{1},\ldots,\alpha_{n}\}

where $n=\dim V$ and each $\alpha\in\mathfrak{h}^{*}$ is written in this list $\dim V_{\alpha}$ — the multiplicity of $\alpha$ — times.

Suppose that $(\sigma,W)$ is another representation of $\mathfrak{g}$ with multiset of weights

\{\beta_{1},\ldots,\beta_{m}\}.

Theorem 7.16.

Suppose that $V,W,\alpha_{i},\beta_{j}$ are as above. Then:

1.

The weights of $V^{*}$ are $\{-\alpha_{1},\ldots,-\alpha_{n}\}$ .
2.

The weights of $V\otimes W$ are

$\{\alpha_{i}+\beta_{j}:1\leq i\leq n,1\leq j\leq m\}.$
3.

The weights of $\operatorname{Sym}^{k}(V)$ are

$\{\alpha_{i_{1}}+\ldots+\alpha_{i_{k}}:1\leq i_{1}\leq i_{2}\leq\ldots\leq i_{% k}\leq n\}.$
4.

The weights of $\Lambda^{k}(V)$ are

$\{\alpha_{i_{1}}+\ldots+\alpha_{i_{k}}:1\leq i_{1}<i_{2}<\ldots<i_{k}\leq n\}.$

Proof.

Let $v_{1},\ldots,v_{n}$ be a basis of weight vectors of $V$ such that $v_{i}$ has weight $\alpha_{i}$ , and let $w_{1},\ldots,w_{m}$ be similar for $W$ with weights $\beta_{i}$ .

1.

The dual basis $v_{1}^{*},\ldots,v_{n}^{*}$ is a basis of weight vectors in $V^{*}$ with $v_{i}^{*}$ having weight $-\alpha_{i}$ .
2.

If $v$ has weight $\alpha$ and $w$ has weight $\beta$ , then for all $H\in\mathfrak{h}$ ,

$\displaystyle H(v\otimes w)$ $\displaystyle=(Hv\otimes w)+v\otimes(Hw)$

$\displaystyle=\alpha(H)v\otimes w+v\otimes\beta(H)w$

$\displaystyle=(\alpha+\beta)(H)(v\otimes w).$

So $\{v_{i}\otimes w_{j}\}$ is a basis of $V\otimes W$ with the given weights.
3.

Similarly to the previous part,

$\{v_{i_{1}}v_{i_{2}}\ldots v_{i_{k}}:1\leq i_{1}\leq i_{2}\leq\ldots\leq i_{k}% \leq n\}$

is a basis of $\operatorname{Sym}^{k}V$ with the given weights.
4.

Similar.

∎

Remark 7.17.

Similar considerations apply to representations of $\mathfrak{sl}_{2,\mathbb{C}}$ (or $\mathfrak{sl}_{n,\mathbb{C}}$ ).

	$\displaystyle H(v\otimes w)$	$\displaystyle=(Hv\otimes w)+v\otimes(Hw)$
		$\displaystyle=\alpha(H)v\otimes w+v\otimes\beta(H)w$
		$\displaystyle=(\alpha+\beta)(H)(v\otimes w).$