**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 met Mary, from Dundee in Scotland, while studying at St Andrews; she was also an undergraduate and later went on to publish translations of French novels by authors such as Zola. Frederick continued his education in Paris and Cologne and married Mary 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 in 1930 at Bedford Park School, 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 Dorothy and Isobel (a fifth child, Francis, died when aged less than two).

It was in Cambridge that Sandy's secondary schooling took place. After one year, 1935-36, at Preparatory to Perse School in Cambridge, Sandy was a foundation scholar at Perse School from 1936 to 1942. He wrote [5]:-

He developed a great interest for science and, since his father was a professor, he [1]:-This school was very good, with a particularly strong science department.

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]:-... always expected to have an academic career.

When Green began his studies at St Andrews, supported by a prestigious Harkness scholarship, he was only sixteen years old so he was too young to serve in the armed 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, taking courses on Mathematics, Physics, Chemistry, and Astronomy, graduating with a B.Sc. in 1944. Then later that year, when he was eighteen years old, he put his university career on hold when he went to Bletchley Park to undertake war work [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.

It was at Bletchley Park that Green first met Margaret Lord, daughter of Herbert and Elizabeth E Lord, whom he eventually married on 2 August 1950 at Girton, Cambridge. She was in the Women's branch of the Royal Navy and had been posted to Bletchley. Sandy and Margaret Green had three children: Jane Margaret Green (who became an epidemiologist working at the Cancer Epidemiology Unit at Oxford University), Sally Ann Green (who became a senior administrator at the University of Sheffield), and Alastair James Green (who became a computer scientist working as Chief Technology Officer of a software company and later as a technology director in a bank in London). Green spent the year 1944-45 at Bletchley Park, then spent the following year at the Royal Aircraft Establishment at Farnborough.... I arrived in August1944, 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.

In 1946, after doing some school teaching, Green returned to the University of St Andrews to complete his first degree. Among his lecturers at St Andrews were Herbert Turnbull, Dan Rutherford and Walter Ledermann. 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 St John's College, University of Cambridge, to undertake research. He wrote [5]:-

He had three Ph.D. thesis advisors in succession, Dudley Ernest Littlewood, Philip Hall and David Rees, and, after submitting his thesisAt Cambridge I was much impressed by lectures given by the algebraist Philip Hall; and in a different way by D E Littlewood.

*Abstract Algebra and Semigroups*, Green was awarded a Ph.D. in 1951. [Sandy's brother Christopher 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 first got to know at Bletchley Park),

*On semi-groups in which x*. In this paper the authors prove that

^{r}= x*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]:-

In 1955 Green publishedMy 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 until1961, I have always regarded him as one of my teachers.

*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:-

It is also highlighted in the citation for the De Morgan Medal which Green received from the London Mathematical Society in 2001:-His remarkable treatment of the irreducible characters of the general linear groups published in1955was 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.

In 1963 Green was appointed as a Reader at the University of Sussex. He was only there for two years but during this time he published perhaps his best-known workIn a1955paper 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 Lusztig fully extended it to the general finite group of Lie type.

*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]:-

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

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 Christopher Zeeman. I [EFR] was fortunate enough to attend an M.Sc. course given by Sandy Green in 1966 at Warwick on 'Representations of groups'. It was an excellent course, one of several outstanding courses I took there including those by Roger Carter on "Group Theory" and David Epstein on "Homological Algebra". The first year Warwick operated there was a "Topology year" and during that year preparations were made for the next year which was a "Group Theory year". Sadly, however, Sandy Green saw little of that year since he suffered a major stroke and took quite a while to regain his health [4]:-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.

Once he had recovered sufficiently so that he could prepare to work again, he read Nathan Jacobson's... he realised he had to look after himself and rest when necessary, and he had immense support from his wife.

*Lie Algebras*in order to get himself back up to pace. His research continued to produce results of great importance and also of great beauty. He changed direction somewhat in the mid 1970s [3]:-

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:-After1975the emphasis of his work changed to algebraic groups and his1981Springer 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 his1955paper.

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:-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.

While mentioning the honours given to Green, we should record that he was elected a fellow of the Royal Society of Edinburgh in 1968, and a fellow of the Royal Society of London in 1987. As another little personal note, let me [EFR] say how delighted I was to be congratulated by Sandy when I myself was elected to the Royal Society of Edinburgh in 1997.The De Morgan Medal for2001was 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 on23November2001. 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,15November2002. 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 July2001. 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!

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.

You can see the whole of this preface at THIS LINK.

One of Sandy Green's achievements was his outstanding role as a Ph.D. supervisor. Many of his students have done an impressive job been carrying on his legacy.

We list some of these students at THIS LINK.

When asked to give his views on educations, Green made the following response [5]:-

We also mention the lectures given by Sandy Green at Groups St Andrews 1989 when he was a main speaker giving a series of lectures onI think that mathematical education in British universities, though generally very good, should contain more(compulsory)theoretical physics. Physics has stimulated the development of much pure mathematics in the past, and in recent years mathematics has again been closely involved in great advancements in theoretical physics. Some of the best recent research in these fields has been done by mathematicians from the former Soviet Union, where it seems that mathematicians learn much more theoretical physics in their undergraduate and postgraduate courses than they would here.

*Schur algebras and general linear groups*. The Proceedings contains a beautiful account of these lectures. Other published lectures include

*Classical invariants*and

*Classical groups*both 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:-

Outside mathematics and his family, Green had a number of interests such as gardening, but the most important was his passion for French literature of 16It is internationally agreed that Green is one of those who have most shaped modern representation theory, and he enjoys widespread respect and affection.

^{th}to the 20

^{th}centuries. The author of [4] writes:-

Above all, Green should be described as a "gentle man". His voice was never raised; logic and clarity sufficed. His lectures were a model of elegance and precision, with a delivery reminiscent of Brauer's. But while he was a mathematician first, he was never aggressively so, and maintained a balance of interests, whether French literature, gardening or his family, and he brought the same degree of interest and care to all those whom he met professionally, whether faculty or students.

**Article by:** *J J O'Connor* and *E F Robertson*

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