Let $V$ be a vector space over a field $k$. Consider the natural action of $SL(V)$ on Sym$^2 V$. Is there a easy formulation of Hilbert Mumford Criterion for semi-stablity of the points of Sym$^2 V$ under this action of $SL(V)$?

For example there is an easy formulation of the semi-stability of the points of $Gr(r,n)$ under the action of $SL(n)$. This is in Newstead's Book "introduction to moduli problem".

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    $\begingroup$ Any element of $\mathrm{Sym}^2V$ can be written $e_1^2+\ldots +e_r^2$ for some linearly independent vectors $e_1,\ldots e_r$. It is immediate to apply the criterion, and see that it is semi-stable if and only if $r=\dim V$. $\endgroup$ – abx Aug 15 '17 at 14:51
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    $\begingroup$ @abx: your characterisation only works if $k$ is algebraically closed with $\operatorname{char} k \neq 2$. (In characteristic $2$, one should distinguish between the symmetric power and the symmetric tensors; I don't know which one the OP has in mind.) $\endgroup$ – R. van Dobben de Bruyn Aug 15 '17 at 18:20
  • $\begingroup$ @abx every element of Sym$^2 V$ corresponds to a vector space map from $V^*\rightarrow V$ ...so the semistable points of Sym$^2 V$ are the elements which gives isomorphism $V^*\rightarrow V$. I wonder whether this holds for $V\otimes V$ also?? i am in char 0 $\endgroup$ – user100841 Aug 15 '17 at 21:57

Here is my attempt for a $d$-dimensional $V$ over an algebraically closed field, like $\mathbb{C}$. Let's identify $\text{Sym}^2(V)$ with the space of all symmetric, contravariant, rank-$2$ tensors or identically the space of all degree $2$ homogeneous polynomials in $d$ variables $(x_1, \cdots , x_d) \equiv \mathbf{x}$. Moreover, we know that every $1$-parameter subgroup $\lambda: \mathbb{C}^* \rightarrow G$ of $G=SL(V)$ is conjugate to \begin{equation} \mathbb{C}^* \ni t \mapsto \lambda(t) = \begin{bmatrix} t^{a_1} & & \\ & \ddots & \\ & & t^{a_d} \\ \end{bmatrix}, \end{equation} for some integers $a_i \in \mathbb{Z}$, such that $a_1 \geq \cdots \geq a_d$ and $\sum_{i=1}^da_i = 0$. According to Hilbert-Mumford criterion, a polynomial $f (\mathbf{x})= \sum_{i,j}c_{ij}x_ix_j$ is unstable if and only if there exists a 1-parameter subgroup $\lambda$ of $SL(V)$ such that $\lim_{t\rightarrow 0} \lambda(t) \, . \, f (\mathbf{x}) = 0$, where \begin{equation} \lambda(t) \, . \, f (\mathbf{x})= f(\lambda^{-1}(t) \, . \, \mathbf{x}) =\sum_{i,j} c_{ij} t^{-a_i - a_j}x_ix_j, \end{equation} which then implies that $f$ in unstable if and only if there exists a $\lambda$ of $SL(V)$ such that $a_i + a_j <0$ for every term in the sum. This means that $f$ can be destabilized ($\lambda(t) \, . \, f$ tends to $0$ as $t \rightarrow 0$). For generalizations to other fields, you may want to take a look at the definition (4.3) or remark (4.5) here.


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