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Schur functions and group representations. (English) Zbl 0483.20006

Astérisque 87-88, 7-19 (1981).
The author’s introduction: “I have been asked to present a report on the history of the Schur calculus. It would have been nice if I had been able to give a complete bird’s eye view of all its facets, stemming from a unified conception from which all its properties could be derived in a natural and logical way. However, extensive reading of the literature has convinced me that no such presentation exists today; perhaps after all it is fortunate for the younger generation that such a challenge remains and may lead to new unsuspected viewpoints on algebra and combinatorics. I have the feeling that we dont’t understand at all the extraordinary interplay of problems should be warmly welcomed, but I feel I am far too ignorant to talk at length on the combinatorial aspects, of which so many specialists in this meeting are very well aware. It is therefore the aspects of the Schur calculus in algebra and analysis which I will try to describe in this talk. What I would like to emphasize is the variety of seemingly unrelated contexts in which the Schur calculus comes up, standing so to speak at the confluence of very different types of theories. If we are some day to reach a unified conception, I believe it will be by a study in depth of the relations between these various theories; to concentrate on a single aspect and deliberately neglect the others is probably a bad policy only likely to lead to dead ends.”
The titles of the sections are: 1. The classical period (up to 1900). 2. Schur’s dissertation and semi-simple algebras. 3. The general theory of linear representations of semisimple Lie groups. 4. Witt vectors and X-rings.
[For the entire collection see Zbl 0468.00006.]

MSC:

20C30 Representations of finite symmetric groups
20C32 Representations of infinite symmetric groups
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 0468.00006
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