Tensor products of coherent configurations
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A Cartesian decomposition of a coherent configuration 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 X comes from a certain Cartesian decomposition. It is proved that if the coherent configuration X is thick, then there is a unique maximal Cartesian decomposition of X, i.e., there is exactly one internal tensor decomposition of 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.
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