Symplectic transformations in n variables with a commutative coefficient field whose characteristic differs from 2 are investigated and invariants are established for the classes of conjugate elements. As in the case of the group of all non-singular linear transformations these invariants consist of irreducible polynomials and systems of non-negative integral numbers, but apart from these also equivalence classes of hermitian forms and of quadratic forms have to be included.Springer had begun publishing mathematical papers well before undertaking the work for his doctoral thesis, having published papers with Nicolaas de Bruijn such as On the zeros of a polynomial and of its derivative (1947) and On the zeros of composition-polynomials (1947). He had also published the paper On induced group characters in 1948 which was reviewed by Richard Brauer who wrote:-
It was proved by the reviewer that every character of a group G of finite order is a linear combination with integral rational coefficients of characters of G induced by linear characters of subgroups. The author gives some simplifications of the proof.After the award of his doctorate, Springer went to France, spending the academic year 1951-52 at the University of Nancy. While there he proved a conjecture of Ernst Witt made in 1937, namely:
As research interests, Springer gives algebra and the theory of linear algebraic groups. He also list more recent research interests as: symmetric varieties and their compactifications, Hecke algebras, complex reflection groups. We mention the four important books which Springer has written. The first was Jordan algebras and algebraic groups (1973). Kevin McCrimmon begins a long review of this book as follows:-
In his review of the book "Jordan-Algebren" by H Braun and M Koecher (1966), the present author took those authors to task for not sufficiently emphasizing the structure group as a linear algebraic group. The present book is designed to remedy that flaw. Throughout it is influenced by the work of Braun and Koecher, and many of the novel ideas in this book emphasize or make explicit results implicit in Koecher's work. It goes over much the same ground, but in the opposite direction: instead of starting with an algebra and seeing how its inverse encodes the algebraic information, the author starts simply from an inversion map satisfying the Hua identity and shows how to decode an algebraic structure from it.Near the end of his review McCrimmon writes:-
For the associatively-inclined this book expunges the dread word "nonassociative" from Jordan theory, since there is nothing nonassociative about inversion; most importantly, it makes Jordan structure theory accessible to the growing audience of persons familiar with root systems. By placing the classification of Jordan algebras in the perspective of classification of certain root systems, the book demonstrates that the structure theories of associative, Lie, and Jordan algebras are not separate creations, but rather instances of the one all-encompassing miracle of root systems.Springer's next book was Invariant theory (1977). V L Popov writes:-
These notes had their origin in a course in invariant theory, given at the University of Utrecht in the autumn of 1975. The purpose of the course was to give an introduction to invariant theory on an elementary level, illustrated by some examples from 19th century invariant theory.Robert B Gardner writes in a review:-
The notes are an enjoyable, readable account of the invariant theory of reductive algebraic groups, concentrating on delicate finiteness theorems. The general theory is illustrated by a detailed analysis of SL(2, K) and finite groups. In particular, the above mentioned theorems of Molien and Chevalley-Serre are clearly presented and lead to interesting explicit calculations for classical reflection groups. I especially recommend these notes to any mathematician who wonders why finiteness theorems are important and how concepts like the integrality of extensions and Noether normalization arose historically. The author has included many references and notes at the end of each chapter, indicating where various results first occurred and why they're significant.In 1981 Springer published Linear algebraic groups which was an expanded version of lectures on linear algebraic groups which he gave at the University of Notre Dame in the autumn of 1978. He writes in the Introduction:-
These notes contain an introduction to the theory of linear algebraic groups over an algebraically closed ground field. They lead in a straightforward manner to the basic results about reductive groups. ... The main difference from the existing introductory texts on this subject ... lies in the treatment of the prerequisites from algebraic geometry and commutative algebra. These texts assume a number of such prerequisites, whereas we have tried to give proofs of everything. We have also tried to limit as much as possible the commutative algebra.The fourth of Springer's books, Octonions, Jordan algebras and exceptional groups (2000), was also based on lectures given by Springer, but these were given in 1963 in German at the University of Göttingen. It was written in collaboration with Ferdinand Veldkamp.
As we have seen from the descriptions of the above books, some have been the result of lecture courses given by Springer at different institutions. Over the years he has made many research visits and we give a list of the main ones:
University of Göttingen (1963);
Institut des Hautes Études Scientifiques, Bures-sur-Yvette (1964, 1973, 1975, 1983);
University of California, Los Angeles (1965-1966);
Tata Institute of Fundamental Research, Bombay (1968, 1980);
University of Paris VII (1971);
University of Warwick (1973);
University of Notre Dame (1978);
Australian National University, Canberra (1981, 1989, 1991, 1997);
University of Paris VI (1984, 1987);
University of Rome II (Tor Vergata) (1986);
University of Basel (1993);
University of Sydney, Australia (1995, 1998, 2001);
Erwin-Schrödinger-Institute for Theoretical Physics, Vienna (2000).
In 1976, Tonny Springer discovered the remarkable fact that the permutation group acts naturally on (the cohomology of) a collection of algebraic varieties, now called Springer fibers. Indeed, all of the irreducible representations - the building blocks of an arbitrary representation - can be constructed by examining the permutation action on a handful of these Springer fibers. Springer's original construction was completely algebraic but was followed by intense activity on the part of many people to give more intrinsically geometric explanations for these representations.
Article by: J J O'Connor and E F Robertson