Exercise 30, § 21

Let F be an algebraically closed field of characteristic $p >0$, let $\mathbf{F}_p$ be the prime subfield of F, and let $G =\mathbf{S}\mathbf{L}_2(\mathbf{F}_p)$. Denote by V the F-vector space $F\otimes_{\mathbf{F}_p}\mathbf{F}^2_p$ endowed with the action of G deduced from the action on $\mathbf{F}^2_p$.

a) Prove that the F[G]-module $\mathsf{S}^d(V)$ is simple for $0\leqslant d < p$. (Let W be a nonzero F[G]-submodule of $\mathsf{S}^d$(V), and let $(e, f)$ be a basis of V. Prove that $e^d$ belongs to W using the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, and then that W is equal to $\mathsf{S}^d(V)$ using the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.) b) Prove that the F[G]-module $\mathsf{S}^p(V)$ is not simple (consider the image of V by the mapping $v\mapsto v^p$).

c) For $p\geqslant 7$, prove that the dimension of $\mathsf{S}^{p-3}(V)$ does not divide the order of G. d) Prove that every simple F[G]-module is isomorphic to one of the modules $\mathsf{S}^d(V)$ for $0\leqslant d < p$ (use Exercise 27, f)).