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.[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.

Isotypic decomposition

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Let   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.,  . For example,   is multiplicity-free as  -module. More precisely, given a closed subgroup H of G, define

 

by setting   and then extending   by linearity. 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  -submodules of  . We can assume  . Let   be the linear functional of W such that  . Then  . That is, the image of   contains   and the opposite inclusion holds since   is equivariant.

Examples

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  • Let   be a B-eigenvector and X the closure of the orbit  . It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.

The Kostant–Rallis situation

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See also

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Notes

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  1. ^ G is not assumed to be connected so that the results apply to finite groups.
  2. ^ Goodman & Wallach 2009, Remark 12.2.2.

References

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  • Goodman, Roe; Wallach, Nolan R. (2009). Symmetry, Representations, and Invariants (in German). doi:10.1007/978-0-387-79852-3. ISBN 978-0-387-79852-3. OCLC 699068818.