Gorsky–Negut rational Shuffle Conjecture for Hikita polynomials

Let (m,n)(m,n) be a coprime pair of positive integers, let Parkm,n\operatorname{Park}_{m,n} be the set of rational parking functions, and let Hm,n[x;q,t]H_{m,n}[\mathbf{x};q,t] be the associated Hikita polynomial. Let Qm,n\mathbf{Q}_{m,n} be the corresponding symmetric-function operator.

Gorsky–Negut rational Shuffle Conjecture. For all coprime pairs of positive integers (m,n)(m,n),

Qm,n(1)n=Hm,n[x;q,t].\mathbf{Q}_{m,n}\cdot(-\mathbf{1})^n=H_{m,n}[\mathbf{x};q,t].

This is the rational analogue of the Shuffle Conjecture, relating the operator construction to Hikita polynomials. No resolution evidence is supplied in the text.

Sources & referencesView supporting material

Primary source

Francois Bergeron, Adriano Garsia, Emily Leven and Guoce Xin, “Compositional (km,kn)-Shuffle Conjectures”, arXiv:1404.4616 (2014).

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