invariant theory

== English == === Noun === invariant theory (countable and uncountable, plural invariant theories) (algebra, representation theory) The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions. 1993, Bernd Sturmfels, Introduction, David Hilbert, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Cambridge University Press, 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 {\displaystyle V} be a K {\displaystyle K} -vector space on which a group G {\displaystyle G} acts linearly. In the ring of polynomial functions K [ V ] {\displaystyle K[V]} consider the subring K [ V ] G {\displaystyle K[V]^{G}} consisting of all polynomial functions on V {\displaystyle V} which are invariant under the action of the group G {\displaystyle G} . The basic problem is to describe the invariant ring K [ V ] G {\displaystyle K[V]^{G}} . In particular, we would like to know whether K [ V ] G {\displaystyle K[V]^{G}} is finitely generated as a K {\displaystyle K} -algebra and, if so, to give an algorithm for computing generators. Used other than figuratively or idiomatically: see invariant,‎ theory. ==== Derived terms ==== geometric invariant theory ==== Translations ==== === Further reading === Geometric invariant theory on Wikipedia.Wikipedia Gram's theorem on Wikipedia.Wikipedia invariant (mathematics) on Wikipedia.Wikipedia Invariant of a binary form on Wikipedia.Wikipedia Representation theory of finite groups on Wikipedia.Wikipedia Symmetry on Wikipedia.Wikipedia invariant theory on nLab