Bell inequalities from group actions of single-generator groups

We study a method of generating Bell inequalities by using group actions of single-generator Abelian groups. Two parties, Alice and Bob, each make one of $M$ possible measurements on a system, with each measurement having $K$ possible outcomes. The probabilities for the outcomes of these measurements are $P(a_j=k,b_{j^{\prime }}=k^{\prime })$, where $j,j^{\prime }\in \{1,2,...,M\}$ and $k,k^{\prime }\in \{0,1,...,K-1\}$. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including $M=2$, $K=3$, and general $M$, $K=2$. We also discuss the resulting inequalities in terms of nonlocal games.