Journal of Combinatorics

Volume 7 (2016)

Number 4

Some remarkable new plethystic operators in the theory of Macdonald polynomials

Pages: 671 – 714



Francois Bergeron (Département de Mathématiques, Lacim, Université du Quebec à Montréal, Canada)

Adriano Garsia (Department of Mathematics, University of California at San Diego)

Emily Sergel Leven (Department of Mathematics, University of California at San Diego)

Guoce Xin (School of mathematical Science, Capital Normal University, Beijing, China)


In the 90’s a collection of plethystic operators were introduced in [2], [7] and [8] to solve some representation-theoretical problems arising from the theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of algebraic combinatorics. Yet, these developments have led to a variety of new conjectures in [3] in the combinatorics and symmetric function theory of Macdonald polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.

Published 16 August 2016