What is invariant theory, really?

Abstract

Invariant theory is the great romantic story of mathematics. For one hundred and fifty years, from its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon, Sylvester and Salmon, and later, Alfred Young, Aitken, Littlewood and Turnbull. In Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in Italy, Capelli, d’Ovidio, Brioschi, Trudi and Corrado Segre; in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics.