Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

Abstract

We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1 \(U_q (\widehat{\mathfrak{s}\mathfrak{l}_2 })\) quantum Knizhnik-Zamolodchikov equation at arbitrary values of the quantum parameter q. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.