invariant theory

Example Sentences

• "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."