Gorsky–Negut rational Shuffle Conjecture for Hikita polynomials

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Let (m,n)(m,n) be a coprime pair of positive integers, let Park⁡m,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.

References

Primary source

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

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