Invariant polynomial

In mathematics, an invariant polynomial is a polynomial in a vector space (see Symmetric algebra), which is based on the vector space is invariant under the action of a group, which

Satisfied for all.

Invariant polynomials in linear algebra

Be a body and the vector space of all matrices over. The general linear group acts on by conjugation:

Invariant polynomials in this case are functions for all.

Examples are the trace and the determinant of matrices. More generally one can ( with a formal variable) the development

Consider and receives invariant polynomials. (The trace and the determinant. Falls is algebraically closed, then generally the k-th elementary symmetric polynomial in the eigenvalues ​​of. )

Invariant polynomials in the theory of Lie groups

Let be a Lie group and its Lie algebra her. A polynomial is a polynomial on ( with real coefficients ) in the basis vectors of, see Symmetric algebra.

The group acts on itself by conjugation: for all. The differential of a linear map

This defines the so-called adjoint representation of the group on the vector space.

An invariant polynomial is a polynomial which is invariant under the adjoint action, ie

Met. The algebra of invariant polynomials is denoted by.

Example

In this case and for. For is the homogeneous polynomial of degree, its value to you as coefficients in the polynomial of degree

Gets for all. ( The values ​​for the set is a polynomial already clearly established. ) The polynomial is called the -th Pontryagin polynomial.

The algebra of the invariant polynomials is generated by the.

Example

For valid, from which first and so then for all odd follows.

The algebra of the invariant polynomials is generated by the.

Example

If is even, one has in addition the Pfaffian determinant, which is defined for with by

The algebra of invariant polynomials is and of the Pontryagin polynomials - Pfaffian determinant of the ( also referred to as Euler polynomial ) generated - if even.

Example

For is the complex-valued homogeneous polynomial of degree, its value to you as coefficients in the polynomial of degree

Gets for all. The polynomial is called the -th Chern polynomial. The Chern and Pontryagin polynomials are related by the equation.

The algebra of the complex-valued invariant polynomials is generated by the.

Example

For is, thus, so are the Chern polynomials on real -valued.

The algebra of the invariant polynomials is generated by the.