A construction of equivariant bundles on the space of symmetric forms

We construct stable vector bundles on the space $mathbb{P}(S^d mathbb{C}^{n+1})$ of symmetric forms of degree $d$ in $n+1$ variables which are equivariant for the action of $ ext{SL}_{n+1}(mathbb{C})$, and admit an equivariant free resolution of length $2$. For $n=1$, we obtain new examples of stable vector bundles of rank $d-1$ on $mathbb{P}^d$, which are moreover equivariant for $ ext{SL}_2(mathbb{C})$. The presentation matrix of these bundles attains Westwick's upper bound for the dimension of vector spaces of matrices of constant rank and fixed size.