Symmetric powers of Nat SL(2,K)
Résumé
In this paper, we identify the representations ${\mathbb{K}[X^{k},X^{k-1}Y,\dots,Y^{k}]}$ among abstract ${\mathbb{Z}[\operatorname{SL}_{2}(\mathbb{K})]}$ -modules. One result is on ${\mathbb{Q}[\operatorname{SL}_{2}(\mathbb{Z})]}$ -modules of nilpotence length at most 5 and generalises a classical "quadratic" theorem by Smith and Timmesfeld; there are reasons to believe that this generalisation is close to optimal. Another result 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.