Incollection,
$A_k$ generalization of the $O(1)$ loop model on a cylinder
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
We define and study the $A_k$ generalization of the $O(1)$
loop model on a cylinder.
This model is a new hybrid generalization of the $O(1)$ loop model:
defined by the affine Hecke algebra and with cylindric
boundary conditions.
First, we introduce a new class of the affine Hecke algebra which
is characterized by the cylindric relations.
We consider the spin representation of the affine Hecke algebra.
The affine Hecke generator is obtained by twisting
the standard Hecke generator by a diagonal matrix.
Second, we consider the representation by the states of the
$A_k$ generalized model.
For this purpose, we introduce a novel graphical depiction,
rhombus tiling with an integer on its face,
to deal with the Yang-Baxter equation and $q$-symmetrizers.
A state of the model is characterized by a path and
constructed through the correspondence
among an unrestricted path, a rhombus tiling and a word.
The cylindric relations become clear by using the rhombus tiling.
Finally, we solve the quantum Kniznik-Zamoldchikov (qKZ)
equation at the Razumov-Stroganov point.
This solution is identified with a special solution of the
qKZ equation of the level $1+1k-k$ constructed
from a non-symmetric Macdonald polynomial. The sum rule
for this model is written as the product of $k$ Schur
functions.
