A group is said to be diagonalizable if it can be represented in a certain way with respect to its action on a vector space, particularly in the context of linear algebra. More specifically, in the context of linear representations, a group is diagonalizable when its representation can be expressed in a diagonal form. In this context, consider a group \( G \) acting on a vector space \( V \) over some field, typically the complex numbers.
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A diagonalizable algebraic group is isomorphic to a closed subgroup of a product of copies of . Every rational representation of such a group decomposes into weight spaces.