  Let $\mathbb{F}$ be a field and let $E$ be the natural representation of
$\mathrm{SL}_2(\mathbb{F})$. Given a vector space $V$, let
$\Delta^{(2,1^{N-1})}V$ be the kernel of the multiplication map $\bigwedge^N V
\otimes V \rightarrow \bigwedge^{N+1}V$. We construct an explicit
$\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes
\bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong \Delta^{(2,1^{N-1})} \mathrm{Sym}^d
E$. This $\mathrm{SL}_2(\mathbb{F})$-isomorphism is a modular lift of the
$q$-binomial identity $q^{\frac{N(N-1)}{2}}[N]_q \binom{d+1}{N+1}_q =
s_{(2,1^{N-1})}(1,q,\ldots, q^d)$, where $s_{(2,1^{N-1})}$ is the Schur
function for the partition $(2,1^{N-1})$. This identity, which follows from our
main theorem, implies the existence of an isomorphism when $\mathbb{F}$ is the
field of complex numbers but it is notable, and not typical of the general
case, that there is an explicit isomorphism defined in a uniform way for any
field.
