  We associate to each synchronous game an algebra whose representations
determine if the game has a perfect deterministic strategy, perfect quantum
strategy or one of several other perfect strategies. when applied to the graph
coloring game, this leads to characterizations in terms of properties of an
algebra of various quantum chromatic numbers that have been studied in the
literature. This allows us to develop a correspondence between various
chromatic numbers of a graph and ideals in this algebra which can then be
approached via various Grobner basis methods.
