Loop models have been extensively studied since the observation by Razumov and Stroganov (2000). Many research areas such as combinatroics, representation theory and integrable systems make contact each other.

In the talk, stating from the basic facts about quantum integrable systems, we analyze the O(1) loop model associated with the Temperley-Lieb algebra. We also consider the "loop" model associated with the Hecke algebra of type A. We mainly discuss the relation to the spin chain model and the ground state wave function.