P-matrices in orbit spaces and invariant theory

TL;DR

This paper reviews how P-matrices, derived from integrity bases, are used to characterize orbit spaces and their strata, aiding in the study of invariant functions under symmetry group actions.

Contribution

It demonstrates that P-matrices are effective tools in constructive invariant theory, especially for determining integrity bases when only partial information is available.

Findings

P-matrices help define equations and inequalities of orbit spaces.

Calculating P-matrix elements can fully determine the integrity basis.

P-matrices are useful even with incomplete integrity basis data.

Abstract

In many physical problems or applications one has to study functions that are invariant under the action of a symmetry group G and this is best done in the orbit space of G if one knows the equations and inequalities defining the orbit space and its strata. It is reviewed how the P-matrix is defined in terms of an integrity basis and how it can be used to determine the equations and inequalities defining the orbit space and its strata. It is shown that the P-matrix is a useful tool of constructive invariant theory, in fact, when the integrity basis is only partially known, calculating the P-matrix elements, one is able to determine the integrity basis completely.