Bethe roots and refined enumeration of alternating-sign matrices

Abstract

The properties of the most probable ground state candidate for theXXZ 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 thestate considered, divided by the maximal component, coincide with the elementary symmetricpolynomials in the corresponding Bethe roots. It is proved that these polynomials are equalto the numbers providing the refined enumeration of the alternating-sign matrices of orderM+1 divided by the total number of the alternating-sign matrices of orderM, for the chainof length 2M+1.