In the 90s a collection of Plethystic operators were introduced in [3], [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 resul
ts 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since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished.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 [2] 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 It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.
In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side
and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime and k > 1.
