]> BibSonomy publications for /bibtex/2e676715358c4e65500abcd8deeb441f2/statphys23 http://www.bibsonomy.org/burst/bibtex/2e676715358c4e65500abcd8deeb441f2/statphys23 BibSonomy BuRST Feed for /bibtex/2e676715358c4e65500abcd8deeb441f2/statphys23 2008-08-21T12:50:46+02:00 $A_k$ generalization of the $O(1)$ loop model on a cylinder http://www.bibsonomy.org/bibtex/2e676715358c4e65500abcd8deeb441f2/statphys23 statphys23 2007-06-20T10:16:09+02:00 affine algebra equation hecke loop model qkz statphys23 topic-1
$A_k$ generalization of the $O(1)$ loop model on a cylinder
K. Shigechi Abstract Book of the XXIII IUPAP International Conference on Statistical Physics (2007)
to affine algebra equation hecke loop model qkz statphys23 topic-1 on 2007-06-20 10:16:09 ]]> Genova, ItalyAbstract Book of the XXIII IUPAP International Conference on Statistical Physics9-13 July$A_k$ generalization of the $O(1)$ loop model on a cylinder2007affine algebra equation hecke loop model qkz statphys23 topic-1 2007-06-20 10:16:09.0We 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 $1+\frac{1}{k}-k$ constructed from a non-symmetric Macdonald polynomial. The sum rule for this model is written as the product of $k$ Schur functions.