Compositional (km,kn)-Shuffle Conjectures

Abstract

In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. 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.

References in corpus (3)

• Rational parking functions and Catalan numbers
• Affine Springer fibers of type A and combinatorics of diagonal coinvariants
• Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials

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• A new Plethystic Symmetric Function Operator and The rational Compositional Shuffle Conjecture at t=1/q
• Interlaced rectangular parking functions
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions