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.
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We establish curious Lefschetz property for generic character varieties of Riemann surfaces conjectured by Hausel, Letellier and Rodriguez-Villegas. Our main tool applies directly in the case when there is at least one puncture where the local monodromy has distinct eigenvalues. We pass to a vector bundle over the character variety, which is then decomposed into cells, which look like vector bundles over varieties associated to braids by Shende-Treumann-Zaslow. These varieties are in turn decomposed into cells that look like C * d−2k × C k . The curious Lefschetz property is shown to hold on each cell, and therefore holds for the character variety. To deduce the general case, we introduce a fictitious puncture with trivial monodromy, and show that the cohomology of the character variety where one puncture has trivial monodromy is isomorphic to the sign component of the S n action on the cohomology for the character variety where trivial monodromy is replaced by regular semisimple monodromy. This involves an argument with the Grothendieck-Springer sheaf, and analysis of how the cohomology of the character variety varies when the eigenvalues are moved around.
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