invariant theory

Noun

 * 1)  The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.
 * 2) * 1993,, Introduction, , Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, , 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.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.

Translations

 * French: théorie des invariants