The quantum symmetric XXZ chain at ∆ = − 12 , alternating sign matrices and plane partitions

We consider the groundstate wavefunction of the quantum symmetric antifer-romagnetic XXZ chain with open and twisted boundary conditions at ∆ = − 1 2 , along with the groundstate wavefunction of the corresponding O(n) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunc-tion components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes. Very recently Razumov and Stroganov [1] have made some remarkable conjectures in which the number of n × n alternating sign matrices appear in certain properties of the ground state wavefunction of the antiferromagnetic XXZ Heisenberg chain at ∆ = −