A proof of the shuffle conjecture

Carlsson, Erik^1, 2 ; Mellit, Anton^3, 4

¹ International Centre for Theoretical Physics, Str. Costiera, 11, 34151 Trieste, Italy
² Department of Mathematics, University of California, Davis, 1 Shields Ave., Davis, California 95616
³ International Centre for Theoretical Physics, Str. Costiera, 11, 34151 Trieste, Italy
⁴ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Journal of the American Mathematical Society, Tome 31 (2018) no. 3, pp. 661-697

Résumé

We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. 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 $\operatorname {Sym}[X]$ over $\mathbb {Q}(q,t)$. We then extend these operators to an action of an algebra $\tilde {\mathbb A}$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.

Détail

DOI : 10.1090/jams/893

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