The Integrable Chiral Potts Model: Quantum Group Methods Applied to Superintegrable Case

The integrable chiral Potts model resulted in the 1980s from a search of new solutions of the star-triangle (Yang--Baxter)\break equations for spin models with expected parafermionic excitations. Its structure relates to cyclic representations of quantum groups at roots of unity, while the so-called superintegrable subcase has additional Onsager algebra structure. Recently the authors have utilized this quantum algebraic information, to derive detailed explicit results for the eigenvectors in the ground state sectors and give new information for the eigenvectors in general. One result is the explicit derivation of the spontaneous magnetization without hidden assumptions, as both conjectures made earlier have now been proved. The explicit eigenvectors also lead to some results for correlation functions. We shall present a review of what has been done so far and discuss the current status of the research.\\ \\ Helen Au-Yang and Jacques H.H. Perk, J. Phys. A: Math. Theor. {\bf 41}, 275201 (2008); {\bf 42}, 375208 (2009); {\bf 43} (2010) 025203 (2010); {\bf 44} 025205 (2011); {\bf 44}, 445005 (2011); arXiv:1108.4713; arXiv:1210.5803.