A new modular plethystic ${\mathrm{SL}}_2(\mathbb{F})$-isomorphism ${\mathrm{Sym}}^{N-1}E\otimes {\bigwedge }^{N+1}{\mathrm{Sym}}^{d+1}E\cong \hat{I}{”}^{(2,1^{N-1})}{\mathrm{Sym}}^{d}E$

Abstract

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 $\hat{I}{”}^{(2,1^{N-1})}V$ be the kernel of the multiplication map ${\bigwedge }^{N}V\otimes V\to {\bigwedge }^{N+1}V$. We construct an explicit ${\mathrm{SL}}_2(\mathbb{F})$-isomorphism ${\mathrm{Sym}}^{N-1}E\otimes {\bigwedge }^{N+1}{\mathrm{Sym}}^{d+1}E\cong \hat{I}{”}^{(2,1^{N-1})}{\mathrm{Sym}}^{d}E$. This ${\mathrm{SL}}_2(\mathbb{F})$-isomorphism is a modular lift of the $q$-binomial identity $q^{\frac{N(N-1)}{2}}[N{]}_q{\left(\genfrac{}{}{0ex}{}{d+1}{N+1}\right)}_q=s_{(2,1^{N-1})}(1,q,\dots ,q^d)$, 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.

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