Given a Lie group acting on the space of independent (spacetime) and dependent (field) variables, it is proved that a divergence invariant variational problem is equivalent to a strictly invariant variational problem if and only if a certain associated cohomology class in the invariant variational bicomplex vanishes. This result is illustrated by several examples, starting with the free particle Lagrangian that appeared in Emmy Noether’s original paper, and includes derivations of associated conservation laws through application of her First Theorem. The chapter concludes with some speculations as to the role such cohomology classes might play in fundamental physics, based on the construction of suitable invariant Lagrangians.

Abstract

This chapter presents the equivariant method of moving frames for finite-dimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows.

Introduction

According to Akivis [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan [22], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric properties of submanifolds and their invariants under the action of transformation groups. In the 1970s, several researchers, cf. [25, 37, 38, 49], began the process of developing a firm theoretical foundation for the method. The final crucial step [32], is to define a moving frame simply as an equivariant map from the manifold back to the transformation group. All classical moving frames can be reinterpreted in this manner. Moreover, the equivariant approach is completely algorithmic, and applies to very general group actions.

Cartan's normalization construction of a moving frame can be interpreted as the choice of a cross-section to the group orbits. This enables one to algorithmically construct an equivariant moving frame along with a complete systems of invariants through the induced invariantization process. The existence of an equivariant moving frame requires freeness of the underlying group action, i.e., the isotropy subgroup of any single point is trivial.

Of all the available methods for computing invariants and covariants, the most incisive are those based on certain invariant differential operators. The Hessian of a binary form, and the Jacobian of a pair of forms, are just special cases of a general method of constructing covariants known as “transvection”. Not only does this process provide another method for constructing explicit covariants and invariants, but, in fact, every covariant and invariant of any system of binary forms arises using these fundamental operations. These ideas have their origins in Cayley's old theory of “hyperdeterminants”, and culminated in their general formulation by the German school of invariant theorists led by Aronhold, Clebsch, and Gordan. The transvection process is fundamental in the implementation of Gordan's algorithm, for constructing a complete system of independent covariants.

In our presentation, we shall concentrate on the formulation of transvectants based on invariant differential operators or “invariant processes” rather than the more traditional symbolic approach. This point of view has several advantages. First, the formulae for the transvection processes are explicitly given in terms of derivatives of the polynomials, rather than the more shadowy “umbral” or symbolic form. As such, one can immediately apply such invariant processes to more general functions and thereby simultaneously implement an “invariant theory of smooth or analytic functions” without additional work. In fact, the homogeneous polynomials will be seen to play a somewhat anomalous, and computationally more difficult, role in the general theory. Second, the use of differential operators makes clear the fascinating and mostly unexplored connections between classical invariant theory and a wide variety of recent developments in mathematics and mathematical physics, including solitons and integrable systems…

We have now reached the culmination of our brief tour of the most noteworthy features in the invariant theory of binary forms and stand ready to launch into new and less well charted territories. Unfortunately, our space limitations prevent us from realizing this grander aim in any depth, but we do have room in this final chapter for a brief introduction to the invariant theory of homogeneous polynomials in more than two variables (along with their inhomogeneous counterparts). Not surprisingly, almost all of the classical — and not so classical — methods that have been developed for binary forms can be straightforwardly adapted to the multivariate situation. On the other hand, the required computations are considerably more involved, and much less is known about explicit bases for invariants and covariants, solution to equivalence problems, canonical forms, structure of symmetries, and so on.

Given a system of multivariate homogeneous polynomials or, more generally, functions, we may construct a wide variety of invariants and covariants by a combination of interrelated techniques, the most useful being modeled on the transvectant processes in conjunction with symbolic or transform methods. Since many of the tools are straightforward extensions of the bivariate versions that were developed in depth in the preceding chapters, we can pass very quickly through them. A significant complication is that, in addition to classical invariants and covariants, there exist nontrivial “contravariants” and “mixed concomitants”, which, unlike the case of binary forms, cannot be directly obtained by a simple determinantal reweighting. Furthermore, our graph-theoretic methods are not nearly as pleasant to devise or use, and so one must, typically, revert to the more involved (albeit more computer-friendly) algebraic manipulations.

As Lie first discovered, the true power of Lie group theory lies in the ability to work on an infinitesimal level. Each Lie group-theoretic object and property has an infinitesimal counterpart. Passage from the finite to the infinitesimal typically leads to a linearization of the underlying conditions, and hence a significant simplification. Moreover, as long as the group is connected, one can always reconstruct the finite object and its properties from their infinitesimal analogs. The resulting combination of group theory and analysis has proven to be exceptionally potent. A striking example is the Killing-Cartan classification of simple Lie groups, [126, 32, 53], which was established over a century ago and forms a (short) part of many textbooks on the subject, [173, 223]. The corresponding problem for simple finite groups was only recently completed and required the arduous labor of many researchers; moreover, the proof runs to thousands of journal pages, cf. [91].

The infinitesimal counterpart of a Lie group is known as a Lie algebra, an algebraic object of independent interest and wide applicability, ranging from physics to abstract algebra. When the group acts on an open subset of Euclidean space, or, more generally, a smooth manifold, the “infinitesimal generators” in the Lie algebra are realized as first order differential operators acting on smooth functions. Our primary focus will be on the infinitesimal approach to invariants and symmetry, leading to the fundamental differential operators which annihilate the invariants and covariants. Interestingly, Cayley, [41], realized the importance of these operators in the context of binary forms many years before Lie arrived on the scene.

While general transformation groups play a ubiquitous role in geometry, the most important for invariant theoretic purposes are the linear versions (along with their projective counterparts). Linear group actions are referred to as “representations”, so as to distinguish them from the more general nonlinear transformation groups. Representation theory is a vast field of mathematical research, and we shall only have room to survey its most elementary aspects, concentrating on those which are relevant to our subject. More complete treatments can be found in numerous reference texts, including. Representation theory has an amazing range of applications in physics and mathematics, including special function theory, quantum mechanics and particle physics, solid state physics, probability, harmonic and Fourier analysis, number theory, combinatorics, bifurcation theory, and many other fields. A quote of I. M. Gel'fand, “ … all of mathematics is some kind of representation theory …”, is perhaps not as far-fetched as it might initially seem! The representations of a general (even nonlinear) transformation group on the scalar-valued functions defined on the space where the group acts are of particular importance. In classical invariant theory, our original transformation rules for inhomogeneous and homogeneous polynomials are very special cases of the general framework of group representations on function spaces. In this fashion, we are led back to our primary subject, renewed and reinvigorated by an understanding of the requisite mathematical theory.

Representations

For simplicity we shall state the basic concepts in representation theory in the context of real group actions on real vector spaces; the corresponding complex version is an immediate adaptation and it is left to the reader to fill in the details.

Classical invariant theory is the study of the intrinsic or geometrical properties of polynomials. This fascinating and fertile field was brought to life at the beginning of the last century just as the theory of solubility of polynomials was reaching its historical climax. It attained its zenith during the heyday of nineteenth-century mathematics, uniting researchers from many countries in a common purpose, and filling the pages of the foremost mathematical journals of the time. The dramatic and unexpected solution to its most fundamental problem — the finitude of the number of fundamental invariants — propelled the young David Hilbert into the position of the most renowned mathematician of his time. Following a subsequent decline, as more fashionable subjects appeared on the scene, invariant theory sank into obscurity during the middle part of this century, as the abstract approach entirely displaced the computational in pure mathematics. Ironically, though, its indirect influence continued to be felt in group theory and representation theory, while in abstract algebra the three most famous of Hilbert's general theorems — the Basis Theorem, the Syzygy Theorem, and the Nullstellensatz — were all born as lemmas (Hilfsätze) for proving “more important” results in invariant theory! Recent years have witnessed a dramatic resurgence of this venerable subject, with dramatic new applications, ranging from topology and geometry, to physics, continuum mechanics, and computer vision. This has served to motivate the dusting off of the old computational texts, while the rise of computer algebra systems has brought previously infeasible computations within our grasp.

One of the barriers awaiting any serious student of the classical theory is the sheer algebraic complexity of many of the constructions. With a view to rendering these complicated manipulations more manageable, not to mention better visualized, Clifford, began developing a graphical method for the description of the invariants and covariants of binary forms, although he died before publishing his findings to any significant extent. Contemporaneously, Sylvester, unveiled his “algebro-chemical theory”, the aim of which was to apply the methods of classical invariant theory to the then rapidly developing science of molecular chemistry. Sylvester's theory was never taken very seriously by chemists, and not developed any further by mathematicians, and soon succumbed to a perhaps well-deserved death. The present graphical treatment of invariant theory, based on results of C. Shakiban and the author, is closest to the version developed by Kempe, which builds on Clifford's posthumous notes.

Although Sylvester envisioned his theory as the future of chemistry, it is Clifford's graph theory that, with one slight but important modification, could have become an important tool in computational invariant theory. The correct framework for the algebro-chemical theory is to use “linear combinations of” digraphs or “directed molecules” as the fundamental objects. One can ascribe both a graph theoretical as well as a chemical interpretation to these objects; both are useful for motivating the method. The fundamental syzygies then translate into certain operations which can be performed on digraphs or, equivalently, certain allowable reactions which can occur among directed molecules.

Using the previous chapter as our motivational springboard, let us now dive into our chosen subject. Most of the classical literature, and indeed most of the present text, is devoted to the simplest case — that of a binary form or homogeneous polynomial in two variables, along with the inhomogeneous univariate counterpart. In this chapter, we shall introduce many of the fundamental concepts in the invariant theory of binary forms. The ideas will be illustrated by the next two most important cases — that of cubic and quartic polynomials. In each case, we shall exhibit a complete system of invariants, as well as a complete list of canonical forms. These examples will serve to motivate the general definitions of invariants and covariants. The emphasis here is on important particular examples, such as Hessians, resultants, and discriminants, and their role in the classification and geometry of binary forms. These initial constructions bring the basic problem of classifying the invariants into focus, leading to the fundamental Basis Theorem of Hilbert, whose proof will appear in Chapter 9. The chapter concludes with a brief discussion of the algebraic relationships, known as “syzygies”, that exist among the fundamental invariants and covariants.

Binary Forms

In the classical literature, homogeneous polynomials are called forms. The adjectives “binary”, “ternary”, and so on refer to the number of variables that the form depends on.

So far, the methods that have been introduced for analyzing the invariants of binary forms have been essentially algebraic in nature and, apart from being geared towards the group of linear transformations in the plane (or projective transformations on the line), are also, in principle, applicable to quite general group actions. We have not yet exploited the additional analytical structure of such matrix groups, which has no counterpart in the realm of finite or discrete groups. Namely, the group elements depend analytically on parameters, and hence the group comes equipped with a topology — notions of open and closed sets, convergence, and so on — as well as a differentiable structure, enabling one to apply powerful calculus-based tools that are not available in the purely discrete framework. The resulting combination of group theory and analysis has a remarkable synergy, enabling one to effectively and efficiently deal with a wide range of practical group-theoretic issues.

In this chapter, we begin with a brief survey of the basic theory of matrix Lie groups, emphasizing the general linear group as the prototypical example. We shall assume the reader is familiar with the basics of multivariable calculus, and we will be a bit less systematic in our inclusion of proofs of main results. The interested reader can find all of the relevant details in standard texts on Lie groups, e.g., [169, 173, 227]. This chapter includes several innovations that are inspired by new extensions, [69, 70], of the Cartan theory of moving frames, cf. [33, 93, 95], arising in the basic geometry of curves and surfaces.