Born: 26 February 1926 in Rochester, New York, USA

Died: 7 April 2014

J A Green is known as Sandy Green to almost everyone in the world of mathematics. His father was Frederick Charles Green and his mother was Mary Balairdie Gilchrist. Frederick Green, only son of James Green and Jessie Isobel Mathieson, had been born in Aberdeen in Scotland in 1891 and, after being educated at Harris Academy in Dundee, obtained his MA from the University of St Andrews. He continued his education in Paris and Cologne and married Mary, from Dundee in Scotland, in 1916. Frederick and Mary Green lived in Canada from 1921 to 1925 when Frederick taught French literature at the University of Manitoba, then went to Rochester, New York, where Frederick worked at the University of Rochester. Sandy was born while the family lived in New York but, shortly afterwards, the family moved to Toronto when Frederick was appointed to the University there. They lived in Toronto, where Sandy began his schooling, until 1935 when Frederick was appointed Drapers Professor of French at the University of Cambridge in England. Sandy was one of his parents' four children, having a younger brother Christopher (who also went on to become a mathematician) and two sisters.

It was in Cambridge that Sandy's secondary schooling took place. He developed a great interest for science and, since his father was a professor, he [1]:-

... always expected to have an academic career.

However, his favourite subject at school was chemistry, so he applied to the University of St Andrews to study that subject. In choosing St Andrews for his undergraduate studies, he was following in his father's footsteps. However [1]:-

... by the time I arrived at university I had decided that mathematics was my true vocation. Fortunately there was no difficulty in changing my course because the curriculum in the Scottish universities is quite flexible.

When Green began his studies at St Andrews he was only sixteen years old so he was too young to serve in the forces although World War II had by this time been taking place for three years. His father, Frederick Green, had served in World War I so had been mobilised as a Captain in September 1939 at the beginning of the war. Frederick was released in 1940 but reposted for special duty in 1942 when he served as a Major. Sandy spent two years as an undergraduate at St Andrews, then in 1944, when he was eighteen years old, he put his university career on hold when he went to Bletchley Park to undertake war work [1]:-

... I arrived in August 1944, and the war in Europe was in its final phase. By that time M H A Newman's plan to use specially designed electronic computers to assist in the decipherment of the "Fish" series of coded messages was well advanced. I was one of a number of new recruits to Newman's section (which was called the Newmanry"), and our main task was to operate these "Colossus" computers, using well-established routines.

It was at Bletchley Park that Green first met Margaret whom he eventually married. She was in the Women's branch of the Royal Navy and had been posted to Bletchley. In 1946 Green returned to the University of St Andrews to complete his first degree. After one year of study he graduated B.Sc. with First Class Honours in Mathematics in 1947 having taken the compulsory courses of Geometry, Algebra, Analysis, Statics, Dynamics and the optional courses of Special Functions, and Algebra in his final year of study. My [EFR] father-in-law, T B Slebarski, who graduated with an M.A. in Mathematics and Astronomy from St Andrews, was in the same class as Sandy Green. He took the first three of these compulsory mathematics courses and the Special Functions optional mathematics course. He often told me what a brilliant undergraduate his fellow student Sandy Green was.

Green then went to the University of Cambridge to undertake research. His thesis advisor was Philip Hall and, after submitting his thesis Abstract Algebra and Semigroups, Green was awarded a Ph.D. in 1951. [Sandy's brother C D Green graduated with a B.Sc. in Applied Mathematics from St Andrews in 1951.] In his thesis Green introduced fundamental relations in a semigroup when he defined what today are called "Green's relations". These five equivalence relations partition elements in terms of the principal ideals that they generate. He published a paper based on his thesis On the structure of semigroups in the Annals of Mathematics in 1951 where properties of Green's relations were developed. In 1952 he published another fundamental paper on semigroups, this time jointly with David Rees (who he had got to know at Bletchley Park), On semi-groups in which x^r = x. In this paper the authors prove that n-generator semigroups in which every elements satisfies x^r = x are finite if and only if the corresponding Burnside groups B(n, r-1) are finite for all n. Green published two further papers in 1952. One was A duality in abstract algebra in which he investigated universal algebras dual to free algebras (where dual means inverting the direction of each homomorphism, inverting the order of all products of homomorphisms and replacing onto homomorphisms by into isomorphisms). The second was On groups with odd prime-power exponent related to an investigation of Burnside groups.

Green had already started his first job by the time he was awarded his doctorate, for in 1950 he was appointed to the University of Manchester. At Manchester, Green's professor was Max Newman whom he had worked under at Bletchley Park some eight years earlier. He also became a colleague of B H Neumann who had similar algebraic interests as Green. It was Tim Wall, however, whose suggestion set Green off on a new line of research [1]:-

My first teaching appointment was at Manchester, and there I learnt from G E Wall (whom I already knew as fellow-student at Cambridge) about the work of Richard Brauer on modular representations. Although I did not meet Brauer until 1961, I have always regarded him as one of my teachers.

In 1955 Green published The characters of the finite general linear groups in the Transactions of the American Mathematical Society. In the citation for the Senior Berwick Prize which Green was awarded by the London Mathematical Society in 1984 the work of this paper is highlighted:-

His remarkable treatment of the irreducible characters of the general linear groups published in 1955 was the guiding light to the subsequent development of the representation theory of reductive algebraic groups over a finite field. In this paper, Green combined with great ingenuity the Frobenius method of inducing characters of subgroups and Brauer's theory of modular representations with deep combinatorics to construct the irreducible characters. He introduced certain polynomials, now called Green polynomials, which were crucial ingredients in the combinatorics and whose generalisation to algebraic groups has been of fundamental importance.

It is also highlighted in the citation for the De Morgan Medal which Green received from the London Mathematical Society in 2001:-

In a 1955 paper Green startled the world of representation theory by giving the complex character table of GL(n, q) in all generality. This was completely unexpected in view of the very partial information available prior to his work. It was not until almost twenty years after this seminal achievement that the work of Deligne and Lustzig fully extended it to the general finite group of Lie type.

In 1963 Green was appointed to the chair of mathematics at the University of Sussex. He was only there for two years but during this time he published perhaps his best-known work A transfer theorem for modular representations (1964) published in the first volume of the Journal of Algebra. It made a major contribution to the modular representation theory of finite groups and established the now fundamental "Green correspondence". In contrast to Brauer's earlier character theoretic approach, Green placed the emphasis on the underlying modules which is the approach prevalent today [2]:-

In his work on the modular representations of finite groups he introduced a new point of view which emphasised the study of the modules in contrast with R Brauer's earlier development in terms of characters. He introduced the concepts of the vertex and the source of an indecomposable module which have been increasingly important in applications. A correspondence, called the Green correspondence, between indecomposable modules for a group and those of its subgroups, has also proved to be exceedingly useful.

Again this important paper is mentioned in [3]:-

This provided the platform on which Dade gave a complete determination of the structure of blocks with cyclic defect group, and led to Green's own development of an axiomatic representation theory and to the categorical representation theory that has been at the centre of much of the most recent activity in this area.

The reason why Green's stay at Sussex was a relatively short one was that, when the University of Warwick opened in 1965, he was one of a number of outstanding mathematicians attracted there by Zeeman. I [EFR] was fortunate enough to attend an M.Sc. course given by Green in 1966 at Warwick onRepresentations of groups. His research continued to produce results of great importance and also of great beauty. He changed direction somewhat in the mid 1970s [3]:-

After 1975 the emphasis of his work changed to algebraic groups and his 1981 Springer volume on polynomial representations of general linear groups where he exploited the Schur algebra has been enormously influential. More recently, he has made substantial contributions to the study of representations of quantum groups via a relationship with the Hall algebras that he had studied earlier in his 1955 paper.

Green remained at the University of Warwick until 1991 when he retired and was made Professor Emeritus. In the 1998 interview [1] he spoke about retirement:-

I remember being surprised that when Philip Hall retired, he stopped doing mathematics - he said he wanted to pursue his interests in history. In my case I have had no doubts that I want to go on doing mathematics, and being a part of the international mathematical scene, as long as possible. Sometimes I am surprised that I am so busy in retirement. But I should not be surprised; this happens because I wish it to be so. It is important to me to keep in contact with other mathematicians, and so I go to more meetings now than I did when I was teaching. I am still very interested in the development of mathematics.

We have mentioned above the award of the Senior Berwick Prize (1984) and the De Morgan Medal (2001) to Green. The award of the Medal is described as follows:-

The De Morgan Medal for 2001 was awarded to Professor J A (Sandy) Green, Emeritus Professor of the University of Warwick; however, owing to illness he was unable to receive the award at the Annual General Meeting on 23 November 2001. With the active co-operation of the Mathematical Institute of the University of Oxford, it was arranged to present the award at a short LMS Meeting before the Institute's regular Colloquium on Friday, 15 November 2002. Thus the President of the LMS made the presentation to Sandy Green after reading the citation for the award, which had appeared in the LMS Newsletter for July 2001. Happily, several members of Sandy's family were present, including Mrs Margaret Green, one son, two daughters and one granddaughter; Sandy's son (Alistair) acted as family photographer!

Finally let us mention Sandy's little book Sequences and series (1958) whose aim is stated in the Preface:-

This book is intended primarily for students of science and engineering ... More emphasis is laid on the illustration of basic ideas by numerical examples, than on formal proofs.

We also mention the lectures given by Sandy at Groups St Andrews 1989 when he was a main speaker giving a series of lectures on Schur algebras and general linear groups. The Proceedings contains a beautiful account of these lectures. Other published lectures include Classical invariants and Classical groupsboth delivered at the University of Coimbra in 1993, and Hall algebras and quantum groups delivered at the University of Coimbra in March 1994. In [3] Green's outstanding achievements are summarised as follows:-

Articles:

1. An interview, Bull. Internat. Center for Mathematics (June 1998), 5-7.
2. Citation for James Alexander Green, Bull. London Math. Soc. 16 (6) (1984), 655-656.
3. Prizewinners : Citation for James Alexander Green, Bull. London Math. Soc. 34 (5) (2002), 633-640.