A new modular plethystic ${\mathrm{S}\mathrm{L}}_2(\mathbb{F})$-isomorphism ${\mathrm{S}\mathrm{y}\mathrm{m}}^{N-1}E\otimes {\bigwedge }^{N+1}{\mathrm{S}\mathrm{y}\mathrm{m}}^{d+1}E\cong {\mathrm{\Delta }}^{(2,1^{N-1})}{\mathrm{S}\mathrm{y}\mathrm{m}}^{d}E$

Let $\mathbb{F}$ be a field and let $E$ be the natural representation of ${\mathrm{S}\mathrm{L}}_2(\mathbb{F})$. Given a vector space $V$, let ${\mathrm{\Delta }}^{(2,1^{N-1})}V$ be the kernel of the multiplication map ${\bigwedge }^{N}V\otimes V\to {\bigwedge }^{N+1}V$. We construct an explicit ${\mathrm{S}\mathrm{L}}_2(\mathbb{F})$-isomorphism ${\mathrm{S}\mathrm{y}\mathrm{m}}^{N-1}E\otimes {\bigwedge }^{N+1}{\mathrm{S}\mathrm{y}\mathrm{m}}^{d+1}E\cong {\mathrm{\Delta }}^{(2,1^{N-1})}{\mathrm{S}\mathrm{y}\mathrm{m}}^{d}E$. This ${\mathrm{S}\mathrm{L}}_2(\mathbb{F})$-isomorphism is a modular lift of the $q$-binomial identity $q^{\frac{N(N-1)}{2}}[N{]}_q{\left(\genfrac{}{}{0px}{}{d+1}{N+1}\right)}_q=s_{(2,1^{N-1})}(1,q,\dots ,q^d)$, 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.