Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration $${\cal X}$$ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of $${\cal X}$$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $${\cal X}$$ is thick, then there is a unique maximal Cartesian decomposition of $${\cal X}$$ , i.e., there is exactly one internal tensor decomposition of $${\cal X}$$ into indecomposable components. In particular, this implies an analog of the Krull-Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.