Symmetric powers of Nat SL(2,K)

TL;DR

This paper characterizes certain symmetric power representations of SL(2,K) within abstract modules, extending classical results and demonstrating methods using group ring computations and Steinberg relations.

Contribution

It identifies symmetric power modules among abstract SL(2,K)-modules and generalizes classical theorems on modules of short nilpotence length.

Findings

Identification of symmetric power modules within abstract modules.

Generalization of classical quadratic theorems by Smith and Timmesfeld.

Extension of linear structures from prime fields to arbitrary fields.

Abstract

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.