What is Representation on coordinate rings?

In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.

Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic group G. G then acts on the coordinate ring k {\displaystyle k} of X as a left regular representation: = f {\displaystyle =f}. This is a representation of G on the coordinate ring of X.

The most basic case is when X is an affine space and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., the quotient of G by a fixed-point subgroup of an involution.