Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

Abstract

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered.

References in corpus (8)

• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions
• Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
• Temperley-Lieb Stochastic Processes
• Baxterized Solutions of Reflection Equation and Integrable Chain Models
• Inhomogeneous loop models with open boundaries
• Quantum incompressibility and Razumov Stroganov type conjectures
• Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models

Cited by in corpus (9)

• Koornwinder polynomials and the XXZ spin chain
• Exact finite size groundstate of the O(n=1) loop model with open boundaries
• Exact spin quantum Hall current between boundaries of a lattice strip
• Boundary transfer matrices and boundary quantum KZ equations
• Separation of variables for symplectic characters
• Finite-size left-passage probability in percolation
• The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
• Matrix product construction for Koornwinder polynomials and fluctuations of the current in the open ASEP
• The Rotor Model with spectral parameters and enumerations of Alternating Sign Matrices