Covariant (invariant theory)

In invariant theory, a covariant is a polynomial expression formed from variables and coefficients of a form (typically a polynomial) that transforms in a specific way under a group of linear transformations. Unlike an invariant, which remains unchanged under the transformations, a covariant transforms according to a representation of the group, maintaining a defined relationship to the original form.

More precisely, let f(x₁, ..., x_n) be a form (homogeneous polynomial) in n variables with coefficients a₁, a₂, ..., a_m. Consider a group G of linear transformations acting on the variables x_i. A covariant is a polynomial C(x₁, ..., x_n, a₁, ..., a_m) such that when the variables x_i are transformed by an element g of G, and the coefficients a_i are transformed accordingly to reflect the effect of g on f, the polynomial C transforms in a predictable way. Specifically, there exists a character χ(g) of the group G such that C transforms as C → χ(g) C.

The transformation law for a covariant distinguishes it from an invariant. While an invariant remains absolutely unchanged under the transformations (i.e., χ(g) = 1 for all g in G), a covariant transforms by a character of the group. The character describes how the covariant is scaled or otherwise modified by the transformation.

Covariants are essential tools for studying the geometric properties of forms that are preserved under a group of transformations. The study of covariants, alongside invariants, provides insight into the structure and classification of forms. The degree of the covariant refers to its total degree in the variables x_i, and the order of the covariant refers to its degree in the coefficients a_i.