invariant theory

Definitions

1. The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.
   countable, uncountable
2. Used other than figuratively or idiomatically: see invariant, theory.
   countable, uncountable

Equivalents

Examples

“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 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.”

“Invariant theory is the great romantic story of mathematics.[…]In our century, Lie⟳ theory and algebraic geometry, differential algebra and algebraic combinatorics are all offsprings of invariant theory.”

“For a linear algebraic group G and a regular representation (#92;rho,V) of G, the basic problem of invariant theory is to describe⟳ the G-invariant elements (#123;#92;bigotimes#125;ᵏⱽ)ᴳ of the k-fold tensor product for all k.”

“The point⟳ is that, to construct locally invariant theories, new fields have⟳ to be introduced which are referred to as the gauge fields.”

CEFR level