# Action of $\mathfrak{sl}_2(\Bbb{C})$ on $\textrm{Sym}^2 V$

I am reading Fulton and Harris and on page 150, there is the following passage (in the second paragraph) that I don't understand:

"Similarly, a basis for the symmetric square $W = \textrm{Sym}^2V = \textrm{Sym}^2 \Bbb{C}^2$ is given by $\{x^2,xy,y^2\}$ and we have $$\begin{eqnarray*} H(x \cdot x) &=& x \cdot H(x) + H(x) \cdot x = 2 x \cdot x\\ H(x\cdot y) &=& x\cdot H(y) + H(x) \cdot y = 0\\ H(y \cdot y) &=& y \cdot H(y) + H(y) \cdot y = 2y \cdot y\end{eqnarray*}$$ so the representation $W = \Bbb{C}\cdot x^2 \oplus \Bbb{C}\cdot xy \oplus \Bbb{C}\cdot y^2 = W_{-2} \oplus W_0 \otimes W_2$ is the representation $V^{(2)}$ above."

I should say $x,y$ are the standard basis vectors of $\Bbb{C}^2$ and $H$ is the matrix $diag(1,-1)$.

My question: What are $W_0$, $W_{-2}$ and $W_2$ in the last line of that paragraph? Also, is it a typo where they write the tensor product of $W_0$ and $W_2$ instead of direct sum? Furthermore, what is this $V^{(2)}$ mentioned? I have looked through the text and there is mentioned things like $V_\alpha$, but none with superscripts.

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There is no typo. The Lie algebra acts on a tensor product as a derivation (i.e., using the product rule from calculus). So, we have: $H ( v \otimes w ) = H(v) \otimes w + v \otimes H(w)$ –  Bombyx mori Sep 19 '12 at 0:48
$V^{(n)}$ is shorthand for $\mathrm{Sym}^nV$, the $n$th symmetric power. Are the $W_{\#}$'s weight spaces? –  anon Sep 19 '12 at 1:02

(1) For a representation $V$ and a weight $\alpha$, $V_{\alpha}$ is the weight space of weight $\alpha$ (see (11.3) on p. 147). So $W_{-2}, W_0, W_2$ are the weight spaces of $W$. The subscripts denote the weights, while of course $W$ refers to the representation at hand.

(2) Yes, the tensor product should be a direct sum. Interestingly, in my version ('Springer study edition', 2004 copyright) this typo is corrected.

(3) The representation $V^{(n)}$ is defined on the very bottom of p. 149 as the (unique!) irreducible representation of $\mathfrak{sl}_2(\mathbb{C})$ with highest weight $n$, i.e. the $(n+1)$-dimensional representation whose weights are $-n, -n + 2, \dots, n-2, n$. In fact, at this point they haven't really constructed these irreps (other than "use the formulas provided and check that they work"), so the point of the discussion on p. 150 is that $V^{(n)} \cong \text{Sym}^n(V)$ (as representations of $\mathfrak{sl}_2(\mathbb{C})$) where $V = V^{(1)}$ is the standard representation of $\mathfrak{sl}_2(\mathbb{C})$ on $V = \mathbb{C}^2$. This is the "standard" construction of the irreps of $\mathfrak{sl}_2(\mathbb{C})$.

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Dear Michael, thanks for your answer. Do you know where I can buy the 2004 edition please? I believe the one sold on amazon is still the 1991 edition. –  user38268 Sep 20 '12 at 11:04
@BenjaLim: I'm not sure to be honest. I believe I purchased my copy through Amazon, but it was several years ago. I'm surprised they would have stopped printing the more recent edition, but it does appear that they have. FWIW, although it seems they have corrected a typo or two, it sounds like the versions are otherwise identical. In particular, the pagination appears to line up exactly, indicating that there have no been major revisions. –  Michael Joyce Sep 20 '12 at 12:10