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Exercises 11.21 and 11.22 in Fulton's Representation Theory are the following:

Let $V$ be the standard representation of $\mathfrak{sl}_2\mathbb{C}$.

  1. The projection map from $\text{Sym}^2(\text{Sym}^3V)$ to $\text{Sym}^2V$ given by the decomposition $$ \text{Sym}^2(\text{Sym}^3V)\cong \text{Sym}^6V\oplus \text{Sym}^2V $$ may be viewed as a quadratic map from the vector space $\text{Sym}^3V$ to the vector space $\text{Sym}^2V$. Show that it may be given in these terms as the Hessian, that is, by associating to a homogeneous cubic polynomial in two variables the determinant of the $2\times 2$ matrix of its second partials.

  2. This projection map can be viewed as associating to an unordered triple of points $\{p,q,r\}$ in $\mathbb{P}^1$ an unordered pair of points $\{s,t\}\subset \mathbb{P}^1$. Show that this pair of points is the pair of fixed points of the automorphism of $\mathbb{P}^1$ permuting the three points $p$, $q$, and $r$ cyclically.

I am uncertain as to how to go about the problem, as I do not quite see how the Hessian matrix relates to the decomposition, or have any feel for the geometric aspects of the second problem. Would anybody care to explain this problem?

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