Physics – Mathematical Physics
Scientific paper
2001-10-09
J. Math. Phys. 43 (2002), 4135-4146
Physics
Mathematical Physics
11 pages, revtex
Scientific paper
10.1063/1.1487445
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.
Batchelor Murray
de Gier Jan
Mitra Saibal
Nienhuis Bernard
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