I am trying to compute the homogenous coordinate ring of Sym$^2$P$^2$. I know that this is equal to
$k[$P$^2 \times$P$^2]^{S_2}$ where $S_2$ acts on $k[$P$^2 \times$P$^2]$ by simultaneously switching $x$ and $x'$, $y$ and $y'$, and $z$ and $z'$ in the homogenous coordinates $[x:y:z;x':y':z']$. Then I believe that:
$k[$P$^2 \times$P$^2]^{S_2} \simeq k[xx',yy',zz',xy'+yx',xz'+zx',yz'+zy']/($relations between these generators$)$
Is this correct? Or should I first embed P$^2 \times$P$^2$ into a higher dimension projective space and then do something similar.
Edit:
I've also tried first embedding P$^2\times$P$^2$ into P$^8$ with the Segre embedding and have that then
$k[$P$^2 \times$P$^2] \simeq k[xx',xy',xz',yx',yy',yz',zx',zy',zz']/(xy'-yx',xz'-zx',yz'-zy').$
However, I am then unsure how to find the elements of this ring fixed by the $S_2$ action. Clearly $xx', yy', zz', xy'+yx',xz'+zx',yz'+zy'$ are all fixed in $k[xx',xy',xz',yx',yy',yz',zx',zy',zz']$ but I am confused about how the quotient affects this.