G-modules and G-varieties

This lecture explores the relationship between G-modules and G-varieties, demonstrating how a chi-variety can be embedded into a finite-dimensional vector space with a linear action that induces the chi-action. The instructor presents a detailed decomposition of the ring of regular functions and proves the isomorphism between v of d and the symmetric power of v star. The lecture also covers a proposition regarding g-equivariant morphisms between affine t varieties and the regular representation. The main observation highlights that every G-variety can be isomorphically embedded into a G-module, showcasing the versatility and applicability of the regular representation in this context.