Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

Garsia, Adriano M.; Xin, Guoce; Zabrocki, Mike

doi:10.1093/imrn/rnr060

Cited by 27 publications

(56 citation statements)

References 15 publications

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“…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].…”

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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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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“…A desirable approach to proving (1.2) would be to determine a recursive formula for D α pX; q, tq, and interpret the result in terms of some commutation relations for ∇. Indeed, this approach has been applied in some important special cases, see [GH02,GXZ12,Hic12]. In [GXZ12], for instance, the authors devise a recursive formula (Proposition 3.12) to prove the Catalan case of the compositional conjecture, extending the results of [GH02].…”

Section: Introductionmentioning

confidence: 99%

A proof of the shuffle conjecture

Carlsson

Mellit

2017

J. Amer. Math. Soc.

134

230

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We present a proof of the compositional shuffle conjecture [HMZ12], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra [HHL`05b]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space V˚whose degree zero part is the ring of symmetric functions SymrXs over Qpq, tq. We then extend these operators to an action of an algebraÃ acting on this space, and interpret the right generalization of the ∇ using an involution of the algebra which is antilinear with respect to the conjugation pq, tq Þ Ñ pq´1, t´1q.2010 Mathematics Subject Classification. 05E05, 05E10, 05A30. 1 (ii) The involution N commutes with d´, and maps y α to N α . (iii) The restriction of N to V 0 " SymrXs agrees with ∇ composed with a conjugation map which essentially exchanges the B α rX; qs and C α rX; qs. It then becomes clear that these properties imply (1.2).While the compositional shuffle conjecture is clearly our main application, the shuffle conjecture has been further generalized in several remarkable directions such as the rational compositional shuffle conjecture, and relationships to knot invariants, double affine Hecke algebras, and the cohomology of the affine Springer fibers, see [BGLX14,GORS14,GN15,Neg13,Hik14,SV11,SV13]. We hope that future applications to these fascinating topics will be forthcoming.1.1. Acknowledgments. The authors would like to thank François Bergeron, Adriano Garsia, Mark Haiman, Jim Haglund, Fernando Rodriguez-Villegas and Guoce Xin for many valuable discussions on this and related topics. The authors also acknowledge the International Center for Theoretical Physics, Trieste, Italy, at which most of the research for this paper was performed. Erik Carlsson was also supported by the Center for Mathematical Sciences and Applications at Harvard University during some of this period, which he gratefully acknowledges.2. The Compositional shuffle conjecture 2.1. Plethystic operators. A λ-ring is a ring R with a family of ring endomorphisms pp i q iPZ ą0 satisfying p 1 rxs " x, p m rp n rxss " p mn rxs, px P R, m, n P Z ą0 q.

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“…For readers who are not familiar with plethystic notation and other symmetric function ammenities, we refer the reader to [3]. Following Macdonald's section on Integral Forms on page 364 of [10], we have the equality…”

Section: Definitions and Formulationsmentioning

confidence: 99%

“…For another example, we have Since the origin is labeled, we do not include a (0, 0) at the end to get ((0, 4), (3, 0), (1, 3), (3, 1), (3,2), (3, 0), (1, 0)).…”

Section: For Each Labelmentioning

confidence: 99%