Invariant theory

English

Noun

1. Template:Lb The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.
   • 1993, Template:W, Introduction, Template:W, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Template:W, page xi,

     Today, invariant theory is often understood as a branch of representation theory, algebraic geometry, commutative algebra, and algebraic combinatorics. Each of these four disciplines has roots in nineteenth-century invariant theory.Template:...In modern terms, the basic problem of invariant theory can be categorized as follows. Let $V$ be a $K$-vector space on which a group $G$ acts linearly. In the ring of polynomial functions $K[V]$ consider the subring $K[V{]}^G$ consisting of all polynomial functions on $V$ which are invariant under the action of the group $G$. The basic problem is to describe the invariant ring $K[V{]}^G$. In particular, we would like to know whether $K[V{]}^G$ is finitely generated as a $K$-algebra and, if so, to give an algorithm for computing generators.

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Derived terms

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Translations

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Further reading

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• invariant theory on Template:W