In
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
, a branch of algebra, given a group ''G'', a covariant is a ''G''-
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
polynomial map
In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as
:P(v) = \sum_ \lambda_(v) \cdots \lambda_( ...
between
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 ...
s ''V'', ''W'' of ''G''. It is a generalization of a classical convariant, which is a homogeneous
polynomial map
In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as
:P(v) = \sum_ \lambda_(v) \cdots \lambda_( ...
from the space of binary ''m''-forms to the space of binary ''p''-forms (over the complex numbers) that is
-equivariant.
See also
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module of covariants In algebra, given an algebraic group ''G'', a group representation, ''G''-module ''M'' and a ''G''-algebra ''A'', all over a field (mathematics), field ''k'', the module of covariants of type ''M'' is the A^G-module
: (M \otimes_k A)^G.
where -^G ...
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Invariant of a binary form#Terminology
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Transvectant
In mathematical invariant theory, a transvectant is an invariant formed from ''n'' invariants in ''n'' variables using Cayley's Ω process.
Definition
If ''Q''1,...,''Q'n'' are functions of ''n'' variables x = (''x''1,...,''x'n'' ...
- method/process of constructing covariants
References
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{{algebra-stub
category:Invariant theory