invariant theory

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invariant theory

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invariant theory (countable and uncountable, plural invariant theories)

1. (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 ${\displaystyle V}$ be a ${\displaystyle K}$-vector space on which a group ${\displaystyle G}$ acts linearly. In the ring of polynomial functions ${\displaystyle K}$ consider the subring ${\displaystyle K^{G}}$ consisting of all polynomial functions on ${\displaystyle V}$ which are invariant under the action of the group ${\displaystyle G}$. The basic problem is to describe the invariant ring ${\displaystyle K^{G}}$. In particular, we would like to know whether ${\displaystyle K^{G}}$ is finitely generated as a ${\displaystyle K}$-algebra and, if so, to give an algorithm for computing generators.

   • 2001, Gian-Carlo Rota, “What is invariant theory, really?”, in H. Crapo, D. Senato, editors, Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo 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 ${\displaystyle G}$ and a regular representation ${\displaystyle (\rho ,V)}$ of ${\displaystyle G}$, the basic problem of invariant theory is to describe the ${\displaystyle G}$-invariant elements ${\displaystyle ({\bigotimes }^{k}V)^{G}}$ of the ${\displaystyle k}$-fold tensor product for all ${\displaystyle k}$.

2. 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.

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