The Shuffle Algebra Revisited

Neguţ, Andrei

doi:10.1093/imrn/rnt156

Cited by 98 publications

(196 citation statements)

References 9 publications

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“…We note also that expressions like (2.25) and more generally the constant term identities of Section 4 are closely related to identities in shuffle algebras [Neg14].…”

Section: Discussion/conclusionmentioning

confidence: 78%

Quantum Q systems: from cluster algebras to quantum current algebras

Francesco

Kedem

2016

Lett Math Phys

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Abstract. In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra A r , and show that it can be viewed as a quotient of the quantum current algebra U q (n[u, u −1 ]) ⊂ U q ( sl 2 ) in the Drinfeld presentation. Moreover, we find the interpretation of the conserved quantities in terms of Cartan currents at level 0, and the rest of the current algebra, in a non-standard polarization in terms of generators in the quantum cluster algebra.

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“…We note also that expressions like (2.25) and more generally the constant term identities of Section 4 are closely related to identities in shuffle algebras [Neg14].…”

Section: Discussion/conclusionmentioning

confidence: 78%

Quantum Q systems: from cluster algebras to quantum current algebras

Francesco

Kedem

2016

Lett Math Phys

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“…Unfortunately most of this material is written in a language that is nearly inaccessible to most practitioners of Algebraic Combinatorics. We were fortunate that the two young researchers E. Gorsky and A. Negut, in a period of several months, made us aware of some of the contents of the latter publications as well as the results in their papers ( [14], [19] and [20]) in a language we could understand. The present developments are based on these results.…”

Section: Our Compositional (Km Kn)-shuffle Conjecturesmentioning

confidence: 99%

Compositional (km,kn)-Shuffle Conjectures

Bergeron¹,

Garsia

Leven

et al. 2015

Int Math Res Notices

136

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[15]. 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. Contents

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“…Without their efforts at translating their results [11], [17], [18] and results of Schiffmann-Vasserot [20], [21], [19] in a language understandable to us, this writing would not have been possible.…”

Section: Acknowledgmentmentioning

confidence: 99%

“…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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confidence: 99%

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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The Shuffle Algebra Revisited

Cited by 98 publications

References 9 publications

Quantum Q systems: from cluster algebras to quantum current algebras

Quantum Q systems: from cluster algebras to quantum current algebras

Compositional (km,kn)-Shuffle Conjectures

Some remarkable new plethystic operators in the theory of Macdonald polynomials

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