invariant theory
English edit
Noun edit
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 nineteenthcentury invariant theory. […] In modern terms, the basic problem of invariant theory can be categorized as follows. Let be a vector space on which a group acts linearly. In the ring of polynomial functions consider the subring consisting of all polynomial functions on which are invariant under the action of the group . The basic problem is to describe the invariant ring . In particular, we would like to know whether is finitely generated as a algebra and, if so, to give an algorithm for computing generators.
 2001, GianCarlo Rota, “What is invariant theory, really?”, in H. Crapo, D. Senato, editors, Algebraic Combinatorics and Computer Science: A Tribute to GianCarlo Rota, Springer,, page 41:
 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.
 2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 225:
 For a linear algebraic group and a regular representation of , the basic problem of invariant theory is to describe the invariant elements of the fold tensor product for all .
 1993, Bernd Sturmfels, Introduction, David Hilbert, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Cambridge University Press, page xi,
 Used other than figuratively or idiomatically: see invariant, theory.
 2012, M. Chaichian, N. F. Nelipa, Introduction to Gauge Field Theories, Springer, page 4:
 The point is that, to construct locally invariant theories, new fields have to be introduced which are referred to as the gauge fields.
Derived terms edit
Translations edit
branch of algebra concerned with actions of groups on algebraic varieties

Further reading edit
 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