2
B.D. SLEEMAN
As to Henry's research suffice it to say that research gave him great personal
pleasure and satisfaction but he had little personal contact with other researchers,
choosing to publish his results in mostly Scottish journals. The cornerstone upon
which Henry Jack's research was built was the development of analytical methods
of group theory to evaluate certain integrals over matrix spaces. Such integrals arise
in multivariate statistics. These ideas were a recurring theme of Henry'reasearch
and ultimately led to his seminal paper
[10]
on analogues of the Schur functions
for the orthogonal group.
Henry's approach to research is nicely summed up in the following remarks of
Professor A. M. Macbeath(1979).
"Jack's interest in integration spaces of matrices had its origin in a problem
which arose in connection with certain work of C. A. Rogers and myself in the
geometry of numbers. There were certain results, which we were practically certain,
were true but further progress depended on being able to make an estimate of the
order of magnitude of certain integrals as a parameter tended to infinity. We
managed with some difficulty to get rather clumsy estimates which were sufficient
for the purpose we had in mind, but at that time I drew Jack's attention to the
problem, mentioning how much more satisfactory it would be if an exact evaluation
were possible. Within a few weeks Jack produced a complete solution in the case
of the first two or three dimensions and then with later studying He solved the
problem completely, with the use of very elegant combinations of algebraic and
analytical techniques.
Encouraged by this success Jack began to look through the literature and saw
that the techniques which he had applied in that particular case were capable of
very much improving and simplifying much of the work that had been done in
connection with multivariate analysis in statistics. In fact he was able to give much
shorter and elegant derivations of many known results, but I think that quite a
lot of this work of his has remained unpublished. He did also find naturally new
applications of his method, and he has produced in the years between 1959 and
now (1978) some ten papers on various applications of his method. Perhaps the
most interesting of these is one, which has appeared in the Proceedings of the Royal
Society of Edinburgh. (This is paper
[10]
of Jack's publications). This paper is not
only merely an evaluation of an integral, but it relates his integrals to classes of
symmetric polynomial, which are of importance in the theory of representations
of the symmetric group. He has discovered a natural basis for the symmetric
polynomials different from those which have already been described."
It
is fitting to recall the prophetic words of Professor Walter Ledermann(1979)
who wrote of Jack's research :
"His long paper on polynomials associated with partitions has a bearing on the
difficult topic of zonal polynomials and I feel that this particular piece of work is
most likely to have an impact on further research."
Henry was honoured for this work by being awarded the Keith Prize of the
Royal Society of Edinburgh in 1970 and in the same year he was elected a Fellow
of the Royal Society of Edinburgh. Although Henry Jack's last published paper
appeared in 1972 he was by then working on a number of conjectures he had
made concerning the symmetric polynomials introduced in
[10].
His notes on this
important development were carefully written out in the manuscript
[13].
All of
Henry's conjectures have turned out to be true, largely due to the work of Stanley
4 CONTEMPORARY MATHEMATICS
Previous Page Next Page