# 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 &sube; A (d &ge; 0) be the kG-submodule of homogeneous polynomials of degree d in A. It is known (and thus you may assume) that V0, . . . , Vp&minus;1 are a complete set of simple modules over kG. If {d1, . . . , dn} is any partition of p, show that the tensor product Vd1 &otimes;k &bull; &bull; &bull;&otimes;k Vdn (under the diagonal G-action) is not semisimple over kG, unless p = 2and n = 1.
Let d1 + &middot; &middot; &middot; + dn= p, where each di &ge; 1. Then f1 &otimes;&middot; &middot; &middot;&otimes;fn &rarr; f1 &middot; &middot; &middot; fn (fi &isin; Vdi ) defines a kG-surjection Vi1 &otimes;&middot; &middot; &middot;&otimes;Vin &rarr; Vp. If p &gt; 2, then Vi1 &otimes;&middot; &middot; &middot;&otimes;Vin cannot be semisimple over kG, since Vpis not. If p = 2, this argument no longer works. However, if n&gt; 1, the conclusion remains valid; namely, V1 &otimes; 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.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!