Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013


In January 1997, during the special year in combinatorics at MSRI, at a dessert party at Hélène’s house, Gil Kalai, in his usual fashion, began asking very pointed questions about exactly what all the combinatorial representation theorists were doing their research on. After several unsuccessful attempts at giving answers that Gil would find satisfactory, it was decided that some talks should be given in order to explain to other combinatorialists what the specialty is about and what its main questions are.

In the end, Arun gave two talks at MSRI in which he tried to clear up the situation. After the talks several people suggested that it would be helpful if someone would write a survey article containing what had been covered in the two talks and including further interesting details. After some arm twisting it was agreed that Arun and Hélène would write such a paper on combinatorial representation theory. What follows is our attempt to define the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status.

Of course this is wholly impossible. Everybody in the field has their own point of view and their own preferences of questions and answers. Furthermore, there is much too much material in the field to possibly collect it all in a single article (even conceptually). We therefore feel that we must stress the obvious; in this article we give a personal viewpoint on the field while remaining aware that there is much important and beautiful work that we have not been able to mention.

On the other hand, we have tried very hard to give a focused approach and to make something that will be useful to both specialists and non specialists, for understanding what we do, for learning the concepts of the field, and for tracking down history and references. We have chosen to write in an informal style in the hope that this way we can better convey the conceptual aspects of the field. Readers should keep this in mind and refer to the notes and references and the appendices when there are questions about the precision in definitions and statements of results. We have included a table of contents at the end of the paper which should help with navigation. Having made these points, and put in a lot of work, we leave it to you the reader, with the earnest hope that you find it useful.

We would like to thank the many people in residence at the special year 1996-97 in combinatorics at MSRI, for their interest, their suggestions, and for continually encouraging us to explain and write about the things that we enjoy doing. We both are extremely indebted to our graduate advisors, A. Garsia and H. Wenzl, who (already many years ago) introduced us to and taught us this wonderful field.

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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