Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

TL;DR

This paper develops multiresidue formulas for solutions to the quantum Knizhnik-Zamolodchikov equation, linking them to generating functions for symmetric plane partitions and alternating sign matrices, and reduces related conjectures to a single integral identity.

Contribution

It introduces multiresidue formulas for the qKZ equation solutions and connects them to combinatorial objects, simplifying conjectures to a single integral identity.

Findings

Derived multiresidue formulas for qKZ solutions

Connected solutions to generating polynomials for plane partitions

Reduced conjectures to a single integral identity

Abstract

We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco '06] connecting the above 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.