Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

TL;DR

This paper links solutions of the q-deformed Knizhnik--Zamolodchikov equation with discrete Hirota equations, showing they generate weighted plane partitions and relate to symmetric alternating sign matrices.

Contribution

It establishes a novel connection between q-KZ solutions, Hirota equations, and enumerations of punctured plane partitions and symmetric matrices.

Findings

Partial sums of q-KZ solutions are solutions to the discrete Hirota equation.

Generated functions count weighted punctured cyclically symmetric plane partitions.

In special cases, these functions match enumerations of symmetric alternating sign matrices.

Abstract

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik--Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of $\tau^2$-weighted punctured cyclically symmetric transpose complement plane partitions where $\tau=-(q+q^{-1})$. In the cases of no or minimal punctures, we prove that these generating functions coincide with $\tau^2$-enumerations of vertically symmetric alternating sign matrices and modifications thereof.