  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.
