# sl(2)-reps categorifying q-binomials

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 $$[n]!=...[n]$$, 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?

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.