# Professor James Alexander (Sandy) Green

### LMS Obituary

Obituaries Index

Professor James Alexander Green, FRS, FRSE, who was elected a member of the London Mathematical Society on 19 June 1958, died on 7 April 2014, aged 88.

J.A. Green, known as Sandy, did his undergraduate studies at St Andrews, but interrupted in the middle by work at Bletchley Park. Sandy moved to Cambridge for his graduate work, supervised by D.E. Littlewood, Philip Hall and David Rees. From 1950 to 1963 he held a teaching position at the University of Manchester, and then a readership at the University of Sussex for two years. From 1965 until his retirement in 1991 he was a Professor at the University of Warwick. After that he moved to Oxford, and became an associate member of the Department, and had MA status in the University. Until last summer he regularly came to the Department for seminars, or to discuss mathematics. Throughout his career, Sandy had a lot of serious health problems, but thanks to the care and support of his wife, Margaret, and his family, Sandy was able to continue his research.

In his PhD thesis, Sandy worked on semigroups. He introduced fundamental relations, now known as 'Green's relations'. Soon after moving to Manchester, Sandy got interested in representation theory. In 1955 he published his paper The characters of finite general linear groups. This was completely unexpected in view of the very incomplete information available prior to his work Sandy then turned to representations of finite groups over fields of prime characteristic and proved many important results. In particular he introduced new invariants, vertices and sources of indecomposable representations, and developed a fundamental correspondence for representations of a group with representations of its $p$-local subgroups. This 'Green correspondence' has become one of the most important tools of the area.

The monograph Polynomial Representations GLn, published in 1980, introduces what Sandy called 'Schur algebras'. Around this time highest weight modules became objects of central interest in algebraic Lie theory. These can be studied via finite dimensional algebras; and Schur algebras are prototypes for such algebras.

More recently, Sandy was involved in the development of the classical Hall algebra theory. In his 1995 paper, he constructed a comultiplication on Hall algebra of a finite directed quiver, and showed that this can be used to show that the Hall algebra is isomorphic to the positive part of the corresponding quantum group.

Sandy was awarded the Senior Berwick Prize in 1984 and the De Morgan Medal in 2001. He was elected to a Fellow of the Royal Society of Edinburgh in 1968, and to a Fellow of the Royal Society of London in 1987.

Karin Erdmann