Violation of Bell inequalities for arbitrary-dimensional bipartite systems

In this paper, we consider the violation of Bell inequalities for quantum system C-K circle times C-K (integer K >= 2) with group theoretical method. For general M possible measurements, and each measurement with K outcomes, the Bell inequalities based on the choice of two orbits are derived. When the observables are much enough, the quantum bounds are only dependent on M and approximate to the classical bounds. Moreover, the corresponding nonlocal games with two different scenarios are analyzed.