  We formulate a general framework for the study of operator systems arising
from discrete groups. We study in detail the operator system of the free group
on $n$ generators, as well as the operator systems of the free products of
finitely many copies of the two-element group $\mathbb Z_2$. We examine various
tensor products of group operator systems, including the minimal, the maximal,
and the commuting tensor products. We introduce a new tensor product in the
category of operator systems and formulate necessary and sufficient conditions
for its equality to the commuting tensor product in the case of group operator
systems. We express various sets of quantum correlations studied in the
theoretical physics literature in terms of different tensor products of
operator systems of discrete groups. We thus recover earlier results of
Tsirelson and formulate a new approach for the study of quantum correlation
boxes.
