The quantum symmetric<i>XXZ</i>chain at Δ = -½, alternating-sign matrices and plane partitions

Abstract

We consider the groundstate wavefunction of the quantum symmetric\nantiferromagnetic XXZ chain with open and twisted boundary conditions at\n$\\Delta=-{1/2}$, along with the groundstate wavefunction of the corresponding\nO($n$) loop model at $n=1$. Based on exact results for finite-size systems,\nsums involving the wavefunction components, and in some cases the largest\ncomponent itself, are conjectured to be directly related to the total number of\nalternating sign matrices and plane partitions in certain symmetry classes.\n