Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde

On FPL configurations with four sets of nested arches

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function adde

Baxter operators for the quantum sl(3) invariant spin chain

The noncompact homogeneous sl(3) invariant spin chains are considered. We show that the transfer matrix with generic auxiliary space is factorized into the product of three sl(3) invariant commuting operators. These operators satisfy the finite difference equations in the spectral parameters which follow from the structure of the reducible sl(3) modules.Comment: 20 pages, 4 figures, references adde

The eight-vertex model and lattice supersymmetry

We show that the XYZ spin chain along the special line of couplings J_xJ_y+J_xJ_z+J_yJ_z=0 possesses a hidden N=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.Comment: 40 pages, 6 figures, Tik