A proof of the shuffle conjecture

Abstract

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 algebraà 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.

The shuffle conjecture

“…Then the touch composition is We can now state the compositional shuffle conjecture of Haglund, Morse and Zabrocki [16,Conjecture 4.5], which was proved by Carlsson and Mellit [6,Theorem 7.5]. ∇C α = π∈P Fn touch(π)=α t area(π) q dinv(π) F n,ides (π) Observe that proving this would immediately prove the shuffle conjecture since if we sum over all α n, then the left-hand side would yield ∇e n by Equation (3.4) and the right-hand side would lose its touch composition restriction.…”

“…On 25 August 2015 Carlsson and Mellit posted an article on the arXiv [5] titled simply "A proof of the shuffle conjecture", in which they proved the compositional shuffle conjecture, which in turn proved the shuffle conjecture. In their proof they refined the compositional shuffle conjecture yet further and proved this refinement.…”

e-Positivity of vertical strip LLT polynomials

“…In [CM18] these operators are called ∆ uv , but we changed the notation in order to avoid confusion with the Delta operators ∆ f defined on Λ.…”

“…Notice that the T i are invertible operators (see [CM18,Section 4] for an explicit formula of the inverse). For k ≥ 0, we define the operators…”

“…The main result that we are going to need about path operators is the following theorem of Carlsson and Mellit which is implict in [CM18], and made explicit in [GHQR19].…”