7 SL₃ 7.8 Irreducible representations of sl_3,C.8 Problems for Michaelmas

7.9 Proof of theorem 7.28 — nonexaminable

Lemma 7.32.

Let $(\rho,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{3,C}$ . Let $v\in V$ be a highest weight vector of weight $\alpha$ , and let

W=\{\rho(Y_{n})\rho(Y_{n-1})\ldots\rho(Y_{1})v:n\geq 0,Y_{i}\in\{E_{21},E_{32}\}.

Then

1.

$W$ is a subrepresentation of $V$
2.

$W_{\alpha}=\left\langle v\right\rangle$ i.e. $v$ is the unique weight vector in $W$ of weight $\alpha$ , up to scaling.
3.

$W$ is irreducible.

Proof.

Let

W_{n}=\{\rho(Y_{m})\rho(Y_{m-1})\ldots\rho(Y_{1})v:n\geq m\geq 0,Y_{i}\in\{E_{% 21},E_{32}\}.

Then

W=\bigcup_{n=0}^{\infty}W_{n}.

Firstly, it is clear that $\rho(E_{21})$ and $\rho(E_{32})$ take $W_{n}$ to $W_{n+1}$ , and so preserve $W$ . Since

\rho(E_{31})=\rho([E_{32},E_{21}])=[\rho(E_{32}),\rho(E_{21})]

we see that $\rho(E_{31})$ also preserves $W$ .

Secondly, every $\rho(Y_{m})\rho(Y_{m-1})\ldots\rho(Y_{1})v$ is a weight vector (by the fundamental weight calculation) and so an eigenvector for all $\rho(H)$ , $H\in\mathfrak{h}$ . Thus $\rho(H)$ preserves each $W_{n}$ (and hence also $W$ ).

Finally, we show that $\rho(E_{12})$ preserves $W_{n}$ . A similar proof then applies for $\rho(E_{23})$ , and then $\rho(E_{13})$ preserves $W$ by the same argument as for $\rho(E_{31})$ . We prove the statement for $\rho(E_{12})$ by induction on $n$ .

For $n=0$ , $W_{0}=\left\langle v\right\rangle$ . Since $v$ is a highest weight vector, $\rho(E_{12})v=0$ and so $\rho(E_{12})(W_{0})\subset W_{0}$ .

Suppose that the claim is true for $n$ . Consider $w=\rho(Y_{n+1})\ldots\rho(Y_{1})v\in W_{n+1}$ with $Y_{i}\in\{E_{21},E_{32}\}$ . We must show that $\rho(E_{12})w\in W_{n+1}$ . Suppose first that $Y_{n+1}=E_{21}$ . Then, as $[E_{12},E_{21}]=H_{12}\in\mathbb{H}$ , we have

	$\displaystyle\rho(E_{12})w$	$\displaystyle=\rho(E_{12})\rho(E_{21})\rho(Y_{n})\ldots\rho(Y_{1})v$
		$\displaystyle=\rho(E_{21})\rho(E_{12})\rho(Y_{n})\ldots\rho(Y_{1})v+\rho(H_{12% })\rho(Y_{n})\ldots\rho(Y_{1})v$
		$\displaystyle\in\rho(E_{21})\rho(E_{12})W_{n}+\rho(H_{12})W_{n}$
		$\displaystyle\subset\rho(E_{21})W_{n}+W_{n}$
by the induction hypothesis and the fact that $W_{n}$ is preserved by $\mathfrak{h}$
		$\displaystyle\subset W_{n+1}+W_{n}$
		$\displaystyle=W_{n+1},$

as required. The proof in the case $Y_{n+1}=E_{32}$ is similar, using that $[E_{12},E_{32}]=0$ .

2.

Note that if $\beta$ the weight of $\rho(Y_{n})\rho(Y_{m-1})\ldots\rho(Y_{1})v$ , with $n$ and $Y_{i}$ as in the lemma, then a calculation using the fundamental weight calculation, as in the proof of lemma [[ref:lem-hwv-exists]], shows that

$l(\beta)=l(\alpha)-n$

and so $\beta\neq\alpha$ if $n>0$ . Since these vectors span $W$ , $v$ is the unique (up to scalar) weight vector in $W$ of weight $\alpha$ .
3.

Suppose that $W$ is reducible. By complete reducibility (Theorem [[ref:thm-complete-reducible-sl]]) we have

$W=U\oplus U^{\prime}$

for $U,U^{\prime}$ nonzero proper subrepresentations of $W$ . We must have $v=u+u^{\prime}$ for unique $u\in U,u^{\prime}\in U^{\prime}$ . The unicity implies that $u$ and $u^{\prime}$ are both weight vectors of weight $\alpha$ ; by part 2, either $u=0$ or $u^{\prime}=0$ . So without loss of generality $v=u\in U$ . But then all $\rho(Y_{n})\ldots\rho(Y_{1})v\in U$ as $U$ is a subrepresentation, so $W=U$ contradicting that $U$ is a proper subrepresentation.

∎

Remark 7.33.

It follows that $W$ as in Lemma 7.32 is actually the subrepresentation generated by $v$ , that is, the span of all vectors obtained by applying arbitrary elements of $\mathfrak{sl}_{3,\mathbb{C}}$ some number of times. The content of the lemma is then that it suffices to apply only $E_{21}$ and $E_{32}$ .

We are now ready to prove Theorem 7.28:

Proof.

First we show the existence. Let $a,b\in\mathbb{Z}_{\geq 0}$ . Consider

V=\operatorname{Sym}^{a}(\mathbb{C}^{3})\otimes\operatorname{Sym}^{b}((\mathbb% {C}^{3})^{*}).

This has a highest weight vector $v=e_{1}^{a}\otimes(e_{3}^{*})^{b}$ of weight $aL_{1}-bL_{3}$ . Let $W$ be the representation generated by $v$ . Then $W$ is irreducible by 7.32 part 3, and has a highest weight vector $v$ of weight $aL_{1}-bL_{3}$ . Thus we can take $V^{(a,b)}=W$ .

Next we show the uniqueness. Suppose that $V, W$ are two irreducible representations with highest weight vectors $v$ and $w$ , respectively, of weight $aL_{1}-bL_{3}$ . Let $U\subset V\oplus W$ be the representation generated by $u=(v,w)$ . Then $U$ is irreducible by 7.32 part 3. The projection $V\oplus W\to V$ sending $(v^{\prime},w^{\prime})$ to $v^{\prime}$ restricts to a homomorphism $U\to V$ which sends $u$ to $v$ . This is therefore a nonzero homomorphism between irreducible representations, and so must be an isomorphism. Thus $U\cong V$ . Similarly $U\cong W$ , and so $V\cong W$ as required. ∎

In fact, it is possible to give an explicit description of the irreducible representations; see problem 97.

Theorem 7.34.

Let $a,b\geq 0$ . Let $V=\mathbb{C}^{3}$

\phi:\operatorname{Sym}^{a}(V)\otimes\operatorname{Sym}^{b}(V^{*})\to% \operatorname{Sym}^{a-1}(V)\otimes\operatorname{Sym}^{b-1}(V^{*})

be the map

(v_{1}\ldots v_{a})\otimes(\lambda_{1}\ldots\lambda_{b})\mapsto\sum_{i=1}^{a}% \sum_{j=1}^{b}\lambda_{j}(v_{i})(v_{1}\ldots\hat{v}_{i}\ldots v_{a})\otimes(% \lambda_{1}\ldots\hat{\lambda_{j}}\ldots\lambda_{b}).

Then $\phi$ is a surjective $\mathfrak{sl}_{3,\mathbb{C}}$ -homomorphism, and its kernel is the irreducible representation with highest weight $aL_{1}-bL_{3}$ .