Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

Question

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:

The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between them?

The answer was affirmative: let $E=k^2$, then the Wronskian isomorphism gives an identification $\bigwedge^m(S^{m+r-1}(E)) \rightarrow S_m(S^r(E))$ as $\operatorname{SL}(E)$-modules.

(Here, $S_m(V)=(V^{\otimes m})^{S_m}$ and $S^m(V)$ is the symmetric power, as a quotient of $V^{\otimes m}$.)

The recent paper Modular plethystic isomorphisms for two-dimensional linear groups of McDowell–Wildon contains the proof of this over any field.

My question is in a similar vein. Let $\nabla^\lambda$ be the Schur functor of the partition $\lambda$, and consider $\lambda_0=(2,1)$. Then, the $\operatorname{SL}(E)$-character of $\nabla^{\lambda_0}(S^m(E))$ is simply $$ \newcommand\qbinom{\genfrac[]0{}}[2]\qbinom{m+2}3, $$where the brackets indicate that these are quantum numbers and binomials, so for instance $[2]=q+q^{-1}$.

Notice that this holds for any $m$. This suggests the possibility that there exists a natural isomorphism $\nabla^{\lambda_0}(S^m(E))\cong E\otimes V_m$, for some "uniformly defined" $\mathrm{SL}(E)$-module $V_m$ of dimension $\qbinom{m+2}3$.

Is there such a natural isomorphism?

Answer 1

This is not a complete answer, but it strongly suggests where to look. Since $\binom{m+2}{3}_q$ is the $q$-character of $\mathrm{Sym}^3\mathrm{Sym}^{m-1}(E)$, working over the complex numbers we have

$$\nabla^{(2,1)} \bigl(\mathrm{Sym}^m E\bigr) \cong E \otimes \mathrm{Sym}^3 \mathrm{Sym}^{m-1}E $$

as representations of $\mathrm{SL}(E)$. Tensoring on the right-hand side with the determinant extends this to an isomorphism of representations of $\mathrm{GL}(E)$ of common polynomial degree $3m$.

In the spirit of my joint paper with McDowell, I now tried introducing suitable dualities into each side. After some experimentation with computer algebra, the only candidate that remains for a uniformly defined isomorphism is

$$\nabla^{(2,1)}\bigl(\mathrm{Sym}_mE\bigr) \cong E \otimes \mathrm{Sym}^3 \mathrm{Sym}_{m-1}E. $$

This isomorphism exists working with modules for $\mathrm{SL}_2(\mathbb{F}_q)$ when $q=16$, $q=27$ and $q=125$ for $m \le 10$. When $q=27$ and $m=10$, of the $8$ possible modules on either side defined by different combinations of the symmetric power functors, only the proposed pair are isomorphic. By the dual version of the modular Wronskian isomorphism proved in my joint paper, the right-hand side could also be written as $E \otimes \bigwedge^3 \mathrm{Sym}_mE$.