Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries

TL;DR

This paper explores the connections between solutions of the open boundary quantum Knizhnik-Zamolodchikov equation and the enumeration of symmetric plane partitions, proposing new conjectures and relations involving quantum parameters.

Contribution

It introduces new conjectures linking qKZ polynomial solutions with symmetric plane partition enumeration and proposes a Razumov-Stroganov type relation in a specific limit.

Findings

Conjectural relations between qKZ sum rules and plane partition enumeration.

Proposed connection between qKZ solutions and symmetric plane partitions.

A new conjecture relating the qKZ limit to refined symmetric plane partition counts.

Abstract

We propose new conjectures relating sum rules for the polynomial solution of the qKZ equation with open (reflecting) boundaries as a function of the quantum parameter $q$ and the $\tau$-enumeration of Plane Partitions with specific symmetries, with $\tau=-(q+q^{-1})$. We also find a conjectural relation \`a la Razumov-Stroganov between the $\tau\to 0$ limit of the qKZ solution and refined numbers of Totally Symmetric Self Complementary Plane Partitions.