Symmetric powers of Nat SL(2,𝕂)

Abstract

Abstract In this paper, we identify the representations 𝕂 ⁢ [ X k , X k - 1 ⁢ Y , … , Y k ] ${\mathbb{K}[X^{k},X^{k-1}Y,\dots,Y^{k}]}$ among abstract ℤ ⁢ [ SL 2 ⁡ ( 𝕂 ) ] ${\mathbb{Z}[\operatorname{SL}_{2}(\mathbb{K})]}$ -modules. One result is on ℚ ⁢ [ SL 2 ⁡ ( ℤ ) ] ${\mathbb{Q}[\operatorname{SL}_{2}(\mathbb{Z})]}$ -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 𝕂 ${\mathbb{K}}$ . All proofs are by computation in the group ring using the Steinberg relations.