In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an invariant polynomial is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

  that is invariant

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 under a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  acting

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 on a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 . Therefore is a -invariant polynomial if


for all and .

Cases of particular importance are for Γ a finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 (in the theory of Molien series

Molien series

In mathematics, a Molien series is a generating function attached to a linear representation ρ of a group G on a finite-dimensional vector space V. It counts the homogeneous polynomials of a given total degree d that are invariants for G...

, in particular), a compact group

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

, a Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 or algebraic group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

. For a basis-independent definition of 'polynomial' nothing is lost by referring to the symmetric powers of the given linear representation of Γ.