A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule

TL;DR

This paper extends the O(1) loop model to an A_k generalization on a cylinder, exploring its algebraic structure, representations, and sum rules through graphical methods and the q-KZ equation.

Contribution

It introduces a new A_k generalized model, characterizes its affine Hecke algebra with cylindric relations, and connects it to the q-KZ equation at a special point.

Findings

Established two algebraic representations of the model.

Developed a graphical rhombus tiling method for Yang-Baxter and q-symmetrizers.

Derived the sum rule by solving the q-KZ equation at the Razumov-Stroganov point.

Abstract

We study the A_k generalized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the A_k generalized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang-Baxter equation and $q$-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov-Stroganov point.