Tensor Product and Semi-stability: Four Variations on a Theme

Abstract

Chevalley proved that in characteristic 0 the tensor product of semi-simple representations is semi-simple. This result has analogues in rather diverse contexts: three of them are presented here in independent sections, focusing on the differences of the frameworks and the similarities of the proofs. Algebraic groups will play a crucial role, sometimes in unexpected ways. RepresentationsLet G be a group and k a field. In this note, a representation of G is a finite-dimensional k-vector space V together with a group homomorphism ρ : G → GL(V). Semi-simple representationsA representation is said to be:• irreducible if there are exactly two sub-vector spaces of V stable under the action of G: the zero subspace 0 and the whole vector space V. In particular the zero representation is not considered to be irreducible.• semi-simple if it can be decomposed into irreducible ones:there are irreducible sub-representationsThe proof of Chevalley's theorem is a beautiful application of the theory of linear algebraic groups (that is, groups of matrices defined by polynomial equations), even though the group G may not at all be of this form.Indeed, in order to prove theorem 1, one may suppose that the field k is algebraically closed and look at GL(V 1 ), GL(V 2 ) as algebraic groups.Then one can suppose G to be itself a linear algebraic group. For, it suffices to take the Zariski-closureḠ of the image of G in GL(V 1 ) × GL(V 2 ), namely the set of pointssuch that f (x) = 0 for all polynomial functions f vanishing identically on the image of G. The semi-simplicity of the representation V 1 ⊗ k V 2 is equivalent for G andḠ, because it is a condition that can be expressed as the vanishing of some polynomials. Now the theory of linear algebraic groups applies: there is an algebraic subgroup rad u (G) of G called the unipotent radical which is connected, unipotent (meaning that all the eigenvalues of its elements are 1), normal and contains any other subgroup of G with these three properties. The unipotent radical controls the semi-simplicity of the representations of G: Applying the preceding fact, the proof of Chevalley's theorem is easily achieved: since the representations V 1 and V 2 are supposed to be semi-simple, an element g of the unipotent radical of G acts as the identity on V 1 and V 2 . Therefore g operates trivially on V 1 ⊗ k V 2 too.