A new “dinv” arising from the two part case of the shuffle conjecture

Duane, Adrian; Garsia, Adriano M.; Zabrocki, Mike

doi:10.1007/s10801-012-0382-0

Cited by 5 publications

(16 citation statements)

References 14 publications

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“…Now from Theorem 2.1 we derive that 3.3 can hold true if and only if 5) an identity that should be verifiable by computer. From this example it is easy to deduce the following general result.…”

Section: The Negut Operators and The Sss Trickmentioning

confidence: 91%

“…]} λ and satisfies the orthogonality condition 5) where , * denotes a deformation of the Hall scalar product, defined by setting…”

Section: Notation and Auxiliary Identitiesmentioning

confidence: 99%

“…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since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished.…”

Section: Introductionmentioning

confidence: 99%

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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since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished.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 It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.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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Section: The Negut Operators and The Sss Trickmentioning

confidence: 91%

“…]} λ and satisfies the orthogonality condition 5) where , * denotes a deformation of the Hall scalar product, defined by setting…”

Section: Notation and Auxiliary Identitiesmentioning

confidence: 99%

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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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“…. C p k 1 suggested to Duane, Garsia, and Zabrocki (in [8]), that the polynomials h m C p 1 C p 2 . .…”

Section: Some Algebraic Theorems and Conjecturesmentioning

confidence: 98%

“…In [8] they only obtain an algorithmic construction of ndinv based on a recursion satisfied by the polynomial h m C p 1 . .…”

Section: Some Algebraic Theorems and Conjecturesmentioning

confidence: 99%

An explicit formula for ndinv, a new statistic for two-shuffle parking functions

Hicks

Kim

2013

Journal of Combinatorial Theory, Series A

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The New dinv is Not So New

D'Adderio

Iraci²

2019

Electron. J. Combin.

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In [Duane, Garsia, Zabrocki 2013] the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in [Hicks, Kim 2013] a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in [Haglund, Remmel, Wilson 2018]. Moreover, we give also a non-compositional proof of the "ehh" case of the shuffle conjecture (after [Garsia, Xin, Zabrocki 2014]) by bijectively proving a relation with the two part case of the Delta conjecture.

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A new “dinv” arising from the two part case of the shuffle conjecture

Cited by 5 publications

References 14 publications

Some remarkable new plethystic operators in the theory of Macdonald polynomials

Some remarkable new plethystic operators in the theory of Macdonald polynomials

An explicit formula for ndinv, a new statistic for two-shuffle parking functions

The New dinv is Not So New

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