Tensor products of coherent configurations

Gang CHEN; Ilia PONOMARENKO

doi:10.1007/s11464-021-0975-9

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Front. Math. China ›› 2022, Vol. 17 ›› Issue (5) : 829-852. DOI: 10.1007/s11464-021-0975-9

RESEARCH ARTICLE

Tensor products of coherent configurations

Gang CHEN¹ ,
Ilia PONOMARENKO¹^,²^,³

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Abstract

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

Keywords

Coherent configuration / Cartesian decomposition / Krull–Schmidt theorem

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Gang CHEN, Ilia PONOMARENKO. Tensor products of coherent configurations. Front. Math. China, 2022, 17(5): 829‒852 https://doi.org/10.1007/s11464-021-0975-9

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