Tensor products of coherent configurations

TL;DR

This paper explores the relationship between tensor and Cartesian decompositions of coherent configurations, proving uniqueness for thick configurations and providing an efficient algorithm for their decomposition.

Contribution

It establishes a correspondence between tensor and Cartesian decompositions and introduces a polynomial-time algorithm for maximal Cartesian decomposition of thick coherent configurations.

Findings

Every tensor decomposition arises from a Cartesian decomposition.

Unique maximal Cartesian decomposition exists for thick coherent configurations.

A polynomial-time algorithm for finding the maximal Cartesian decomposition is provided.

Abstract

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.