Bethe roots and refined enumeration of alternating-sign matrices

Abstract

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.

arXiv commentLaTeX 2e, 12 pages, minor corrections, references added

References in corpus (13)

• Spin chains and combinatorics
• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions
• The Importance of being Odd
• Spin chains and combinatorics: twisted boundary conditions
• Large distance asymptotic behavior of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain
• Emptiness formation probability of the XXZ spin-1/2 Heisenberg chain at Delta=1/2
• A new way to deal with Izergin-Korepin determinant at root of unity
• Eight-vertex model and non-stationary Lame equation
• Ground State of the Quantum Symmetric Finite Size XXZ Spin Chain with Anisotropy Parameter $\Delta =1/2$
• The six and eight-vertex models revisited
• The XXZ spin chain at $\Delta =-1/2$: Bethe roots, symmetric functions and determinants
• Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
• Finite Size XXZ Spin Chain with Anisotropy Parameter $\Delta =1/2$

Cited by in corpus (8)

• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• Current Fluctuations in the exclusion process and Bethe Ansatz
• Nondiscriminatory Propagation on Trees
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
• A possible combinatorial point for XYZ-spin chain
• Multiply-refined enumeration of alternating sign matrices
• Factorised solutions of Temperley-Lieb $q$KZ equations on a segment
• The nineteen-vertex model and alternating sign matrices