Some remarkable new plethystic operators in the theory of Macdonald polynomials

Abstract

In the 90's 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 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since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished.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 [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 It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion. IntroductionOur main actors in this development are the operators D k introduced in [8], whose action on a symmetric function F [X] is defined by setting Thus A is clearly a graded algebra. What is surprising is that A is in fact bi-graded by simply assigning the generators D k bi-degree (1, k).To make this more precise consider first where Π u,v denotes the portion of Π which is a linear combination of words in D of total bi-degree (u, v). To show that A is bi-graded it is necessary and sufficient to prove that Π, as an operator, acts by zero on Λ if and only if all the Π u,v act by zero. This is one of the very first things we will prove about A.The connection of A to the above mentioned developments is that it gives a concrete realization of a proper subspace of the Elliptic Hall Algebra studied by Schiffmann and Vasserot in [20], [21] and [19]. In particular it contains a distinguished family of operators {Q u,v } of bi-degree given by their index that play a central role in the above mentioned conjectures. For a co-prime bi-degree their construction is so simple that we need only illustrate it in a special case.For instance, to obtain Q 3,5 we start by drawing the 3 × 5 lattice square with its diagonal (the line (0, 0) → (3, 5), as shown in the adjacent figure), we then look for the lattice point (a, b) that is closest to and below the diagonal. In this case (a, b) = (2, 3). This yields the decomposition (3, 5) = (2, 3) + (1, 2) and we setWe must next work precisely in the same way with the 2 × 3 rectangle and, as indicated in the adjacent figure, obtain the decomposition (2, 3) = (1, 1) + (1, 2) and setNow, in this case, we are done, since it turns out that we may setIn particular by combining 0.2, 0.3 and 0.4 we obtainIn the general co-prime case (m, n), the precise definition is based on an elementary number theoretical Lemma that characterizes th...

Compositional (km,kn)-Shuffle Conjectures

“…In the following section we describe how the modular group SL 2 (Z) acts on the operators Q m,n and use this action to justify our definition of the operators Q km,kn . Elementary proofs that justify the uses we make of this action are given in [4]. Here we also show how these operators can be efficiently programmed on the computer.…”

“…For more on the coprime case, see [1]. We will discuss further aspects of the more general case in [4], a paper in preparation. The first object in Figure 3 gives a (7, 9)-parking function and the second object gives a 7 × 9 table of ranks.…”

“…The present developments are based on these results. Nevertheless, for sake of completeness we have put together in [4] a purely Algebraic Combinatorial treatment of all the background needed here with proofs that use only the Macdonald polynomial "tool kit" derived in the 90's in [2], [3], [8] and [9], with some additional identities discovered in [10].…”

“…In [4], SL 2 [Z] is shown to act on the algebra generated by the D k operators by setting, for its generators N := 1 1 0 1 and…”

“…The definition of the Q operators in the non-coprime case, as well as some of their remarkable properties, appear in the following result proved in [4].…”