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The previous answers are very good, and I too found them useful, however I would like to point out another relation in the same vein as Dr. Tao's. Thus, this is not an answer to your question, but something that you may wish to notice if you are thinking along these lines.

This ?short? set of facts is a categorical way to think of representations as fields and the corresponding Galois groups. Notably, we get something that looks like a "Galois Correspondence" which gives an isomorphism between the group and the tensor automorphisms. This is Tannaka-Krein Theorem, which has many nice applications. I attempt here to be brief, but give you the ideas, I apologize for mistakes.

*Proposition*Consider a monoidal categories of k-representations of a compact group G, the category of k-vector-spaces and the functor, F, s.t.

(Rep_k(G), *_k,k^1)---->(Vect_k, *_k, k^1),


where k^1 is the unit object with trivial representation. There exists F which forgets the representation in the following way;

F((V_1,p_1) *_k (V_2,p_2))=  F((V_1,p_1)) *_k F((V_2,p_2))= V_1 *_k V_2.


Now if we consider tensor automorphisms of a functor F

Definition The tensor automorphisms of a functor F in C is the set

Aut^*_C(F)={natural transformations from F to itself preserving * in C.


Lemma There exists a group homomorphism T:G--->Aut^*_c(F) mapping

x|---->x^~ s.t. x^~(p,V_p)=p(x)


which is monomorphic iff for all x there is some p s.t. p(x) is not identity.

Tannaka-Krein Theorem If G is compact, then T is an isomorphism.

This theorem in some sense can be seen as a Galois Correspondence.