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For questions about modular representation theory, the study of representations over a field of positive characteristic.
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How to prove that $M^G=\mathbb{F}_p[x_1\cdot v, x_1^{p\cdot(p-1)}+ v^{p-1}]$?
Let $G=Sl_2(\mathbb{F}_p)$ and $M= \mathbb{F}_p[x_1,x_2]$, where $p$ is a prime.
$M$ is a $G$-module with $(A\cdot x_1, A\cdot x_2)=(x_1,x_2)\cdot A, (\forall) A \in Sl_2(\mathbb{F}_p)$.
I have to …