In mathematics, an invariant polynomial is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

P

that is invariant under a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

\Gamma

acting on a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

V

. Therefore,

P

is a

\Gamma

-invariant polynomial if :

P(\gamma x) = P(x)

for all

\gamma \in \Gamma

and

x \in V

. Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

, a Lie group or

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

. For a basis-independent definition of 'polynomial' nothing is lost by referring to the

symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...

s of the given

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

of Γ.

References

Commutative algebra Invariant theory Polynomials {{algebra-stub