Cherednik's duality conjecture for reduced superpolynomials

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Let Pn,mλ(u,q,t)\mathcal{P}_{n,m}^{\lambda}(u,q,t) be the stabilized superpolynomial and define its reduced version by

Pn,mλ,red(u,q,t)=Pn,mλ(u,q,t)P1,0λ(u,q,t).\mathcal{P}_{n,m}^{\lambda,\mathrm{red}}(u,q,t)=\frac{\mathcal{P}_{n,m}^{\lambda}(u,q,t)}{\mathcal{P}_{1,0}^{\lambda}(u,q,t)}.

Cherednik's duality conjecture. For the transposed partition λt\lambda^t,

q(1−n)∣λ∣Pn,mλt,red(u,q,t)=t(n−1)∣λ∣Pn,mλ,red(u,t−1,q−1).q^{(1-n)|\lambda|}\mathcal{P}_{n,m}^{\lambda^t,\mathrm{red}}(u,q,t)=t^{(n-1)|\lambda|}\mathcal{P}_{n,m}^{\lambda,\mathrm{red}}(u,t^{-1},q^{-1}).

This is the second DAHA conjecture stated in the source; no resolution is supplied there.

References

Primary source

Eugene Gorsky and Andrei Neguţ, “Refined knot invariants and Hilbert schemes”, arXiv:1304.3328 (2015).

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