Answer to Question #18650 in Algebra for Mohammad

Question

For any field k of characteristic p, let G = SL2(Fp) act on the polynomial ring A = k[x, y] by linear changes of the variables {x, y}, and let Vd ⊆ A (d ≥ 0) be the kG-submodule of homogeneous polynomials of degree d in A. It is known (and thus you may assume) that V0, . . . , Vp−1 are a complete set of simple modules over kG. Compute the composition factors of Vp, and show that Vp is semisimple over kG if and only if p = 2 .