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.
If {d1, . . . , dn} is any partition of p, show that the tensor product Vd1 ⊗k • • •⊗k Vdn (under the diagonal G-action) is not semisimple over kG, unless p = 2and n = 1.

Answer 1

Let d1 + · · · + dn= p, where each di ≥ 1. Then f1 ⊗· · ·⊗fn → f1 · · · fn (fi ∈ Vdi ) defines a kG-surjection Vi1 ⊗· · ·⊗Vin → Vp. If p > 2, then Vi1 ⊗· · ·⊗Vin cannot be semisimple over kG, since Vpis not. If p = 2, this argument no longer works. However, if n> 1, the conclusion remains valid; namely, V1 ⊗ V1 is still not semisimple. In fact, for p =2 ,G = SL2(Fp) is the group S3, and V1 is easilyseen to be isomorphic to the module V. Thus, the desired conclusion nowfollows immediately.