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.[1] G then acts on the coordinate ring
of X as a left regular representation:
This is a representation of G on the coordinate ring of X.
The most basic case is when X is an affine space (that is, X is a finite-dimensional representation of G) 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.
be the sum of all G-submodules of
that are isomorphic to the simple module
; it is called the
-isotypic component of
Then there is a direct sum decomposition: where the sum runs over all simple G-modules
The existence of the decomposition follows, for example, from the fact that the group algebra of G is semisimple since G is reductive.
X is called multiplicity-free (or spherical variety[2]) if every irreducible representation of G appears at most one time in the coordinate ring; i.e.,
is multiplicity-free as
More precisely, given a closed subgroup H of G, define by setting
and then extending
The functions in the image of
are usually called matrix coefficients.
Then there is a direct sum decomposition of
-modules (N the normalizer of H) which is an algebraic version of the Peter–Weyl theorem (and in fact the analytic version is an immediate consequence.)
Proof: let W be a simple
We can assume
be the linear functional of W such that
and the opposite inclusion holds since
is equivariant.