Question #23888

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.

Expert's answer

Let *d*1 + *· · · *+ *dn*= *p*, where each *di ≥ *1. Then *f*1 *⊗**· · ·**⊗**fn → f*1 *· · · fn *(*fi **∈** Vdi *) defines a *kG*-surjection *Vi*1 *⊗**· · ·**⊗**Vin → Vp*. If *p > *2, then *Vi*1 *⊗**· · ·**⊗**Vin cannot *be semisimple over *kG*, since *Vp*is not. If *p *= 2, this argument no longer works. However, if *n>* 1, the conclusion remains valid; namely, *V*1 *⊗** V*1 is still not semisimple. In fact, for *p *=2 ,*G *= SL2(F*p*) is the group *S*3, and *V*1 is easilyseen to be isomorphic to the module *V*. Thus, the desired conclusion nowfollows immediately.

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