Compositional (km,kn)-Shuffle Conjectures

Abstract

In 2008, Haglund et al. [21] formulated a Compositional form of the Shuffle Conjecture of Haglund et al. [20]. In very recent work, Gorsky and Negut, by combining their discoveries [19, 25, 26] with the work of Schiffmann and Vasserot [28, 29] on the symmetric function side and the work of Hikita [22] and Gorsky and Mazin [18] on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of [20]. 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 contribution here is a compositional extension of the Gorsky–Negut Shuffle Conjecture for each pair |$(km,kn)$|⁠, with |$(m,n)$| coprime and |$k > 1$|⁠.