Symmetric powers of Nat SL(2,K)

We identify the representations $mathbb{K}[X^k, X^{k-1}Y, dots, Y^k]$ among abstract $mathbb{Z}[mathrm{SL}_2(mathbb{K})]$-modules. One result is on $mathbb{Q}[mathrm{SL}_2(mathbb{Z})]$-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.