Cherednik–Danilenko conjecture for compactified Jacobian Poincaré polynomials

Let (r1,s1),,(r,s)(r_1,s_1),\ldots,(r_{\ell},s_{\ell}) be a sequence of characteristic pairs for an algebraic curve. Starting with f+1:=p1Λf_{\ell+1}:=p_1\in\Lambda, recursively apply the operators γri,si\gamma_{r_i,s_i} and evaluate at 11 as described in the paper, obtaining a symmetric function f1f_1 of degree r1rr_1\cdots r_{\ell}. Cherednik–Danilenko conjecture. The specialization at t=1t=1 of

(f1,er1r)(f_1,e_{r_1\cdots r_{\ell}})

agrees with the Poincaré polynomial of the compactified Jacobian of an algebraic curve with characteristic pairs (r1,s1),,(r,s)(r_1,s_1),\ldots,(r_{\ell},s_{\ell}). This conjecture connects symmetric functions constructed from the characteristic pairs with the geometry of compactified Jacobians; the cited source presents it as a conjecture, and no resolution is supplied here.

Sources & referencesView supporting material

Primary source

Eugene Gorsky, Mikhail Mazin and Alexei Oblomkov, “Generic curves and non-coprime Catalans”, arXiv:2210.12569 (2024).

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