Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves the subspace $k[x_1,x_2,y_1,y_2]_{s,k}$ of polynomials that are homogeneous of degree $s+k$ with total $x_i$ degree $s$ and total $y_i$ degree $k$. I think these are sometimes said to be of bidegree $(s,k)$, but I'm not entirely sure that's standard terminology.

A computation I've performed in a seemingly unrelated mathematical field has led me to believe that for all $d \geq 0$, there should be a nonzero $\operatorname{SL}_2(k)$-invariant polynomial in $k[x_1,x_2,y_1,y_2]_{d,d}$ that is unique up to scaling.

**Question**: Assuming I'm right, how can I go about writing this polynomial down explicitly?