Introduction to Invariant theory

I attempted to write a self-contained introduction to classical invariant theory focused on computation of groundforms. Almost only classical notions are used. If you understand German go to the Download-section and download parts of it. For a very short introduction read the English or German abstract.

The aim of the thesis was to implement effecient pre-Hilbert methods of computing groundforms of binary quantics. The three main algorithms used were

• Gordan's algorithm of transvectants (Überschiebungen)
• Cayley's brute force approach for invariants
• Sylvester's tamisage

• Peter. J. Olver, Graph Theory and Classical Invariant Theory, Advances in Mathematics, 75, 1989, p. 212-245
• Arthur Cayley , A Second Memoir Upon Quantics , Philosophical Transactions of the Royal Society of London, 146, 1856, p. 102-126;
  or Collected Papers, vol. II, p. 251-275
• Fabian Franklin , On the Calculation of the Generating Functions and Tables of Groundforms of Binary Quantics, American Journal of Mathematics, 3, 1880, p. 128-153

If you are interested in a good introduction to classical invariant theory I highly recommend the French article

• Jacques Dixmier, Quelques aspects de la théorie des invariants, Gazette des Mathématiciens, 43, 1990, p. 39-64

• Joseph P. S. Kung & Gian-Carlo Rota, The Invariant Theory of Binary Forms, Bulletin of the American Mathematical Society, Nr. 1, Vol. 10, 1984, p. 27-85

• Wilhelm Franz Meyer, Bericht über den gegenwärtigen Stand der Invariantentheorie, Jahresberichte der Deutschen Mathematiker-Vereinigung, 1892, p. 81-292

• J. H. Grace & A. Young, The Algebra of Invariants, Cambridge, 1903
• Isaai Schur & Helmut Grunsky, Vorlesung über Invariantentheorie, Springer-Verlag, Berlin und Heidelberg, 1968
• Bernd Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Wien, 1993

• Tony Crilly, The Rise of Cayley's Invariant Theory (1841-1862), Historia Mathematica, 13, 1986, p. 241-254
• Tony Crilly, The Decline of Cayley's Invariant Theory (1863-1895), Historia Mathematica, 15, 1988, p. 332-347
• Karen Hunger Parshall, Toward a History of Nineteenth-Century Invariant Theory, in "The History of Modern Mathematics", ed. David E. Rowe & John McCleary, vol. I, Boston, 1989, p. 157-206
• Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Julius Springer Verlag, Berlin, 1926, p. 155-166

Portraits of six important mathematicians with respect to invariant theory can be seen at People.