  A number of conjectures have been given recently concerning the connection
between the antiferromagnetic XXZ spin chain at $\Delta = - \frac12$ and
various symmetry classes of alternating sign matrices. Here we use the
integrability 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 for
the XXZ chain for periodic, twisted and open boundary conditions. These include
expressions for the elementary symmetric functions evaluated at the groundstate
solution of the Bethe roots. In this approach Schur functions play a central
role and enable us to derive determinant expressions which appear in certain
natural double products over the Bethe roots. When evaluated these give rise to
the numbers counting different symmetry classes of alternating sign matrices.
