In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.

1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

The theory of Lie groups and their representations is a vast subject (Bourbaki [Bou] has so far written 9 chapters and 1,200 pages) with an extraordinary range of applications. Some of the greatest mathematicians and physicists of our times have created the tools of the subject that we all use. The appearance of a book on the subject by a well-known researcher is thus noteworthy. In this review I shall discuss briefly the modern development of the subject from its historical beginnings in the mid-nineteenth century and describe how the book by Claudio Procesi fits into the overall picture. The origins of Lie theory are geometric and stem from the view of Felix Klein (1849–1925) that geometry of space is determined by the group of its symmetries. As the notion of space and its geometry evolved from Euclid, Riemann, and Grothendieck to the supersymmetric world of the physicists, the notions of Lie groups and their representations also expanded correspondingly. The most interesting groups are the semisimple ones, and for them the questions have remained the same throughout this long evolution: What is their structure? Where do they