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Person: Green (2), J. A.

Sandy Green was an American born British mathematician who worked in finite groups and representation theory.

Mathematical Profile (Excerpt):

• 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 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.
• 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.
• 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.
• 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.
• Fortunately there was no difficulty in changing my course because the curriculum in the Scottish universities is quite flexible.
• 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.
• 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.
• She was in the Women's branch of the Royal Navy and had been posted to Bletchley.
• Green spent the year 1944-45 at Bletchley Park, then spent the following year at the Royal Aircraft Establishment at Farnborough.
• 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.
• 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 had three Ph.D. thesis advisors in succession, Dudley Ernest Littlewood, Philip Hall and David Rees, and, after submitting his thesis Abstract Algebra and Semigroups, Green was awarded a Ph.D. in 1951.
• 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.
• 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.
• In 1955 Green published The characters of the finite general linear groups in the Transactions of the American Mathematical Society.
• 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.
• 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.
• 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 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".
• 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.
• 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".
• he realised he had to look after himself and rest when necessary, and he had immense support from his wife.
• Once he had recovered sufficiently so that he could prepare to work again, he read Nathan Jacobson's 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.
• 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.
• We have mentioned above the award of the Senior Berwick Prize (1984) and the De Morgan Medal (2001) to Green.
• 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.
• More emphasis is laid on the illustration of basic ideas by numerical examples, than on formal proofs.
• 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.
• 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.
• 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 on 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.
• 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.

Born 26 February 1926, Rochester, New York, USA. Died 7 April 2014, Botley, Oxford, England.

View full biography at MacTutor

Tags relevant for this person:

Group Theory, Origin Usa

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non-Github:

@J-J-O'Connor

@E-F-Robertson

References

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive