Symmetric powers of Nat SL(2,K)

We identify the representations $\mathbb{K}[X^k,X^{k-1}Y,\dots ,Y^k]$ among abstract $\mathbb{Z}[{\mathrm{S}\mathrm{L}}_2(\mathbb{K})]$-modules. One result is on $\mathbb{Q}[{\mathrm{S}\mathrm{L}}_2(\mathbb{Z})]$-modules of short nilpotence length and generalises a classical “quadratic” theorem by Smith and Timmesfeld. Another one is on extending the linear structure on the module from the prime field to $\mathbb{K}$. All proofs are by computation in the group ring using the Steinberg relations.