Compositional (<i>km</i>,<i>kn</i>)-Shuffle Conjectures

Bergeron, François; Garsia, Adriano M.; Leven, Emily; Xin, Guoce

doi:10.1093/imrn/rnv272

Cited by 52 publications

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“…Gorsky and Negut [GN13] show how the results of Aganagic and Shakirov on torus knot invariants can be expressed in terms of Macdonald polynomials using advanced objects such as the Hilbert scheme. Bergeron, Garsia, Leven, and Xin [BGLX14a], [BGLX14b] have shown how this Macdonald polynomial construction can be done combinatorially with plethystic symmetric function operators, and in fact they define operators Q (m,n) for any relatively prime (m, n) by a recursive procedure. The rational shuffle conjecture can then be phrased as Q (m,n) (−1) n = B (m,n) (x 1 , .…”

Section: Rational Catalan Combinatoricsmentioning

confidence: 99%

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Haglund¹

2015

Handbook of Enumerative Combinatorics

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This chapter contains an account of a two-parameter version of the Catalan numbers, and corresponding two-parameter versions of related objects such as parking functions and Schröder paths, which have become important in algebraic combinatorics and other areas of mathematics as well. Although the original motivation for the definition of these objects was the study of Macdonald polynomials and the representation theory of diagonal harmonics, in this account we focus only on the combinatorics associated to their description in terms of lattice paths. Hence this chapter can be read by anyone with a modest background in combinatorics. In Section 1 we include basic facts involving q-analogues, permutation statistics, and symmetric functions which we need in later sections. Sections 2, 3, and 4 contain the results on the q, t-versions of the Catalan numbers, parking functions, and Schröder paths, respectively. Section 5 contains a brief account of the recent exciting extensions of these objects which have arisen in the study of string theory, knot invariants, and the Hilbert scheme from algebraic geometry.

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Section: Rational Catalan Combinatoricsmentioning

confidence: 99%

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Haglund¹

2015

Handbook of Enumerative Combinatorics

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“…The resulting operator, which will be denoted "G km,kn ," turns out to have a variety of surprising properties. In fact, computer exploration led to the discovery (in [2]) that in many instances the symmetric polynomial G km,kn (−1) k(n+1) has a conjectured combinatorial interpretation as an enumerator of certain families of "rational" Parking Functions.One of the most surprising contributions to this branch of Algebraic Combinatorics is a recent deep result [18] of Andrei Negut giving a relatively simple but powerful constant term expression for the action of the operators Q m,n . The reader is referred to the findings concerning the Negut formula that are presented in [2] for the reasons we used the word "powerful" in this context.…”

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“…In this case (a, b) = (2,3). This yields the decomposition (3, 5) = (2, 3) + (1, 2) and we set (0.2)…”

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“…The reader is referred to the findings concerning the Negut formula that are presented in [2] for the reasons we used the word "powerful" in this context. Here it has been one of our priorities to give a straight-forward proof of Negut's formula using only tools developed in the present treatment of the subject.…”

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Some remarkable new plethystic operators in the theory of Macdonald polynomials

Bergeron

Garsia

Leven

et al. 2016

Journal of Combinatorics

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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...

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Combinatorics of the Diagonal Harmonics

Hicks

2019

Association for Women in Mathematics Series

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Compositional (km,kn)-Shuffle Conjectures

Cited by 52 publications

References 35 publications

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Some remarkable new plethystic operators in the theory of Macdonald polynomials

Combinatorics of the Diagonal Harmonics

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