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A number of conjectures have been given recently concerning the connectionbetween the antiferromagnetic XXZ spin chain at $\Delta = - \frac12$ andvarious symmetry classes of alternating sign matrices. Here we use theintegrability of the XXZ chain to gain further insight into these developments.In doing so we obtain a number of new results using Baxter's $Q$ function forthe XXZ chain for periodic, twisted and open boundary conditions. These includeexpressions for the elementary symmetric functions evaluated at the groundstatesolution of the Bethe roots. In this approach Schur functions play a centralrole and enable us to derive determinant expressions which appear in certainnatural double products over the Bethe roots. When evaluated these give rise tothe numbers counting different symmetry classes of alternating sign matrices.

The XXZ spin chain at Δ=−1/2: Bethe roots, symmetric functions, and determinants