sl(2)-reps categorifying q-binomials

Question

Recall that the $q$-binomial coefficient $\big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]$ is the Laurent polynomial in $q$ given by $$ \big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]=\frac{[a]!}{[b]![a-b]!} $$ where $[n]!=[1][2][3]...[n]$, and $[i]=\tfrac{q^i-q^{-i}}{q-q^{-1}}=q^{i-1}+q^{i-3}+q^{i-5}+...+q^{-(i-1)}$.

Let $V_n$ be the $n$-dimensional irrep of $\mathfrak{sl}(2)$.

It is easy to check that there exists an isomorphism of $\mathfrak{sl}(2)$-representations $$ S^k V_{n+1} \cong \textstyle \bigwedge^k V_{n+k} $$ between the $k$-th symmetric power of $V_{n+1}$ and the $k$-th exterior power of $V_{n+k}$, as both have $\big[\begin{smallmatrix}n+k\\k\end{smallmatrix}\big]$ as their character

Proof: We exhibit a basis of weight vectors of $S^k V_{n+1}$, a basis of weight vectors of $\bigwedge^k V_{n+k}$, and a bijection between these bases which respects the weights. Let $x_0,\ldots, x_n$ be the standard basis of $V_{n+1}$, and let $y_1,\ldots y_{n+k}$ be the standard basis of $V_{n+k}$. Then $x_{i_1}{\cdot}\, x_{i_2} \cdot... \cdot x_{i_k}\mapsto y_{i_1+1}\wedge y_{i_2+2}\wedge...\wedge y_{i_k+k}$ is the desired bijection. $\square$

The problem with the above proof is that it only proves the existence of an isomorphism, without actually constructing one.

Question: It there a natural isomorphism of $\mathfrak{sl}(2)$-representations $S^k V_{n+1} \cong \textstyle \bigwedge^k V_{n+k}$ that one can write?

Answer 1

Yes there is. In my article with Chipalkatti "On the Wronskian combinants of binary forms" in J. Pure Appl. Algebra, we gave an explicit construction in Section 2.5. We called it the Wronskian isomorphism because when followed (on the sym-sym side) by complete symetrization, we get the usual Wronskian (in homogenized form). Now, I'm sure the map was known a long time before. The earliest reference I found is the Thesis by Cyparissos Stephanos from 1884. Essentially, the map is nothing more than dividing by the Vandermonde in order to turn antisymmetric functions into symmetric ones, a kind of Boson-Fermion correspondence.

You might be interested in a new development about this map regarding characteristic $p$. See the article "Koszul modules and Green’s conjecture" by Aprodu et al., Sec 3.4.