Transvectant of binary forms
= Transvectant of binary forms
{title2=$(f,g)_r\in\operatorname{Sym}^{m+n-2r}(\mathbb C^2)$}
For forms of degrees $m,n$, the $r$th transvectant contracts $r$ pairs of indices using the invariant alternating form. Explicitly it is a scalar normalization of $\sum_{j=0}^r(-1)^j\binom rj(\partial_x^{r-j}\partial_y^j f)(\partial_x^j\partial_y^{r-j}g)$. These contractions realize the irreducible summands in the symmetric-power tensor-product decomposition.