34 - Invariant Theory: Cayley and Sylvester

Summary

Preliminary Remarks

The invariant theory of forms, with forms defined as homogeneous polynomials in several variables, was developed extensively in the nineteenth century as an important branch of algebra but with very close connections to algebraic geometry. Several ideas and methods of invariant theory were influential in diverse areas of mathematics: topics as concrete as enumerative combinatorics and the theory of partitions and as general as twentieth-century abstract commutative algebra.

George Boole, the highly original British mathematician, may be taken as the founder of invariant theory, though early examples of the use of invariance can be found in the works of Lagrange, Laplace, and Gauss. Boole had almost no formal training in mathematics, but he carefully studied the work of great mathematicians, including Newton, Lagrange, and Laplace. In a paper on analytic geometry written in 1839, Boole took the first tentative steps toward the idea of invariance, but he gave a clearly formulated definition in his 1841 “Exposition of a General Theory of Linear Transformations.” He wrote that he found his inspiration in Lagrange's researches on the rotation of rigid bodies, contained in the 1788 Mécanique analytique. Lagrange's result is most economically described in terms of matrices, a concept developed in the 1850s by Cayley. In modern terms, Lagrange's problem was to diagonalize a 3 × 3 symmetric matrix A; Lagrange expressed this in terms of binary quadratic forms.