Recall that the [$q$-binomial coefficient](https://en.wikipedia.org/wiki/Gaussian_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?
> 
> 
> 



sl(2)-reps categorifying q-binomials
Yes there is. In my article with Chipalkatti ["On the Wronskian combinants of binary forms"](https://www.sciencedirect.com/science/article/pii/S0022404906001812) 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](https://books.google.com/books?id=vXpAsTWGZTQC&printsec=frontcover#v=onepage&q&f=false) 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"](https://link.springer.com/article/10.1007/s00222-019-00894-1) by Aprodu et al., Sec 3.4.


