Sandy Green: Mathematician who worked at Bletchley before becoming a leading figure in the discipline of representation theory

In June 1944 Sandy Green graduated from the University of St Andrews aged 18 with a degree in mathematics, awarded after two years under wartime regulations, and was almost immediately dispatched to Bletchley Park. He was not the youngest mathematician there – that distinction goes to Oliver Atkin – nor quite the last to survive, but surely it shaped his career and personal life, for he was assigned to Block F, headed by the mathematician Max Newman, and it was there, too, that he met his wife, Margaret Lord, who was working on Colossus.

He was born in Rochester, NY. His father, Frederick Charles Green, was a professor, Scottish by birth and outlook. His father was appointed the second Drapers Professor of French at Cambridge in 1935, where Green was educated at the Perse School. For university, the family tradition was St Andrews, and there he was to go, intending to study chemistry but changing to mathematics, returning after the war for an honours degree before moving to Cambridge for his doctorate.

His supervisors were DE Littlewood, Philip Hall and David Rees. Hall was Britain’s leading group theorist but was lecturing on “Abstract algebra”. Green’s thesis was on semigroups, and the first chapter is based on Hall’s lectures. Whether Hall suggested extending that work to semigroups, we do not know; the thesis does not tell us, and Hall himself never wrote any papers on them. Nor is there any indication in the thesis itself that Rees influenced more than a small part. However, he would have met many mathematicians at Bletchley, including Rees, but he could not have said so then. No record exists of what he did there, although he described himself as a “Human computor” (the spelling used at Bletchley for people, to distinguish them from machines). But 60 years on, what are known as the “Green relations” in semigroups have found unexpected applications to the classification of formal languages (such as computer languages) by means of formal relations that define them.

Max Newman appointed him to an assistant lectureship in Manchester in 1950 as he built that department along the lines he had developed at Bletchley. The intensity of a research-oriented mathematics department, which while normal today was virtually unheard of in Britain then, served him well. His 1955 paper on the characters of the general linear groups was seminal, laying the foundations for the work of Lusztig and many others since.

Green was always well read mathematically. In that paper he made heavy use of ideas going back to Frobenius and Schur, but also of the then recent work of Richard Brauer. This led him to examine Brauer’s modular representation theory from a different perspective, that of modules (which in a different context had appeared in his thesis) in particular the significant role played by indecomposable modules where he established the important global-local relationship known as the Green correspondence that led to the widespread study of representations of finite groups via the associated modules in preference to character theory.

In 1963 Green moved to a Readership at the University of Sussex, and two years later to a Chair at Warwick. Sadly, he suffered a major stroke a year after arriving; he realised he had to look after himself and rest when necessary, and he had immense support from his wife. Once he returned to work, its quality was not impaired and he continued to develop new ideas. Formalisation of his module theoretic approach led to the Green ring; later he returned to promulgate ideas that had originated with Philip Hall in his development of what he termed Hall algebras, and this work has had fruitful consequences in its application to quantum groups.

The Greens lived in a Georgian house in Warwick. Often, seminars would be followed by a party there, where he would rest before coming down for dinner. After his retirement in 1991 they moved to Oxford, where he remained an active mathematical figure and regular participant in seminars. Soon he was pressed into the University’s service to join a committee reviewing mathematics. His calm commonsense quietly contributed to recommendations that transformed the Mathematical Institute.

Some of his ideas were revolutionary for Oxford. If statutory chairs were the only way to appoint senior people from outside, such opportunities should be taken, and the University should find alternative ways to promote its own faculty – heady stuff for a time when internal promotions had been suspended on financial grounds, but commonplace today.

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.

Green was elected a Fellow of the Royal Society in 1987, and was awarded the Senior Berwick Prize of the London Mathematical Society in 1984 and in 2001, the centenary of Brauer’s birth, its De Morgan Medal, a fitting tribute to a man whom history should regard as Brauer’s natural successor.

James Alexander Green, mathematician: born Rochester, New York 26 February 1926; married 1950 Margaret Lord (two daughters, one son); died Oxford 7 April 2014.