# Answer to Question #23888 in Abstract Algebra for john.george.milnor

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.

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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