Compositional (km,kn)-Shuffle Conjectures

TL;DR

This paper extends the Compositional Shuffle Conjecture to a broader family of conjectures for pairs (km, kn), building on recent advances linking symmetric functions and combinatorial structures.

Contribution

It introduces a new compositional extension of the Gorsky-Negut Shuffle Conjecture for pairs (km, kn), generalizing previous conjectures for coprime (m,n) and their multiples.

Findings

Established a compositional extension for (km, kn) conjectures

Connected recent symmetric function and combinatorial developments

Provided a framework for further exploration of 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.