Relative Invariants from Moving Frames on an Extended Manifold

TL;DR

This paper introduces a modified moving frame method to construct relative invariants for Lie group actions, linking them to absolute invariants on an extended manifold, with explicit formulas and examples for the projective group.

Contribution

It develops a constructive approach to generate relative invariants using twisted group actions and invariantization, extending the classical moving frame method.

Findings

Established a one-to-one correspondence between relative and absolute invariants.

Provided explicit formulas for relative invariants in terms of fundamental invariants.

Demonstrated the method with examples involving the projective group PGL(3, R).

Abstract

A constructive modification of the moving frame method is developed in this paper for the construction of relative invariants of regular Lie group actions. Let a relative invariant $I$ of weight $\omega$ transform according to the rule $$ I(g \cdot \boldsymbol x) = \mu(g, \boldsymbol x)^{\omega} I(\boldsymbol x), $$ where $\mu: G \times \mathcal{M} \to \mathbb{R}^\times$ is a scalar multiplier (1-cocycle). It is shown that the cocycle property of $\mu$ is equivalent to the well-definedness of the twisted group action on the extended manifold $\widehat{\mathcal{M}} = \mathcal{M} \times \mathbb{R}^\times$, and that relative invariants on $\mathcal{M}$ are in one-to-one correspondence with absolute invariants of this action on $\widehat{\mathcal{M}}$. The main result is that, given a moving frame, the invariantization of the multiplier is a canonical relative invariant of weight $-1$. This enables the constructive realization of any weight and yields an explicit formula for an arbitrary relative invariant in terms of the fundamental absolute invariants and the invariantized multiplier. Examples are provided to demonstrate the application of the proposed approach for the projective group $PGL(3, \mathbb{R})$.