qKZ equation and Alternating Sign Matrices
نویسنده
چکیده
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.
منابع مشابه
Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models
for Ak−1 models with open boundaries, by constructing polynomial solutions of level one boundary quantum Knizhnik–Zamolodchikov equations for Uq(sl(k)). The result takes the form of a character of the symplectic group, that leads to a generalization of the number of vertically symmetric alternating sign matrices. We also investigate the other combinatorial point q = −1, presumably related to th...
متن کاملMore refined enumerations of alternating sign matrices
We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d = 2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers An,i,j that enumerat...
متن کاملFunctorial Properties of the Hypergeometric Map E. Mukhin and A. Varchenko
The quantized Knizhnik-Zamolodchikov equation is a difference equation defined in terms of rational R matrices. We describe all singularities of hypergeometric solutions to the qKZ equations.
متن کاملPattern Avoidance in Alternating Sign Matrices
We generalize the definition of a pattern from permutations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schröder numbers, 1, 2, 6, 22, 90, 394 . . .. We give a bijection between 132-avoiding alternating sign matrices and Schröder-paths, which gives a refined enumeration. We also show that the 132, 123avoiding alternatin...
متن کاملAffine Alternating Sign Matrices
An Alternating sign matrix is a square matrix of 0’s, 1’s, and −1’s in which the sum of the entries in each row or column is 1 and the signs of the nonzero entries in each row or column alternate. This paper attempts to define an analogue to alternating sign matrices which is infinite and periodic. After showing the analogue we define shares desirable cahracteristics with alternating sign matri...
متن کامل