Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.