Tensor products of coherent configurations

Abstract

A Cartesian decomposition of a coherent configuration $\mathcal{X}$ 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 $\mathcal{X}$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $\mathcal{X}$ is thick, then there is a unique maximal Cartesian decomposition of $\mathcal{X}$, i.e., there is exactly one internal tensor decomposition of $\mathcal{X}$ 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.

21 pages