Representation on coordinate rings explained

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

k[X] of X as a left regular representation: (g ⋅ f)(x) = f(g ^{-1} x) . 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.

Isotypic decomposition

Let

k[X] _{(λ)} be the sum of all G-submodules of k[X] that are isomorphic to the simple module V ^{λ} ; it is called the λ -isotypic component of k[X] . Then there is a direct sum decomposition:k[X] = o plus _{λ} k[X] _{(λ)} where the sum runs over all simple G-modules V ^{λ} . 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) if every irreducible representation of G appears at most one time in the coordinate ring; i.e.,

\operatorname{dim} k[X] _{(λ)} \le \operatorname{dim} V ^{λ} .For example, G is multiplicity-free as G x G -module. More precisely, given a closed subgroup H of G, define\phi _{λ} : V ^{{λ} *} ⊗ (V ^{λ} ) ^{H} \to k[G/H] _{(λ)} by setting \phi _{λ} (\alpha ⊗ v)(gH) = \langle \alpha, g ⋅ v \rangle and then extending \phi _{λ} by linearity. The functions in the image of \phi _{λ} are usually called matrix coefficients. Then there is a direct sum decomposition of G x N -modules (N the normalizer of H)k[G/H] = o plus _{λ} \phi _{λ} (V ^{{λ} *} ⊗ (V ^{λ} ) ^{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 G x N -submodules of k[G/H] _{(λ)} . We can assume V ^{λ} = W . Let \delta _{1} be the linear functional of W such that \delta _{1(w)} = w( 1 ) . Then w(gH) = \phi _{λ} (\delta _{1} ⊗ w)(gH) .That is, the image of \phi _{λ} contains k[G/H] _{(λ)} and the opposite inclusion holds since \phi _{λ} is equivariant.

Examples

• Let

v _{λ} \in V ^{λ} be a B-eigenvector and X the closure of the orbit G ⋅ v _{λ} . It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.

See also

• Algebra representation

References

• Book: Goodman . Roe . Wallach . Nolan R. . Nolan R. Wallach . Symmetry, Representations, and Invariants . 2009 . 978-0-387-79852-3 . 10.1007/978-0-387-79852-3 . 699068818 . de.

Notes and References

1. G is not assumed to be connected so that the results apply to finite groups.