Tensor products of coherent configurations

doi:10.1007/s11464-021-0975-9

Front. Math. China

2022, Vol. 17

Issue (5) : 829-852 https://doi.org/10.1007/s11464-021-0975-9

RESEARCH ARTICLE

Tensor products of coherent configurations

Gang CHEN¹(

), Ilia PONOMARENKO^1,^2,³

¹. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
². Steklov Institute of Mathematics at St. Petersburg, Russia
³. Sobolev Institute of Mathematics, Novosibirsk, Russia

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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

Corresponding Author(s): Gang CHEN

Issue Date: 28 December 2022

Cite this article:

Gang CHEN,Ilia PONOMARENKO. Tensor products of coherent configurations[J]. Front. Math. China, 2022, 17(5): 829-852.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0975-9
https://academic.hep.com.cn/fmc/EN/Y2022/V17/I5/829

1	R A Bailey, P J Cameron, C E Praeger, C. Schneider The geometry of diagonal groups. arXiv: 2007.10726
2	G Chen, I. Ponomarenko Coherent Configurations. Wuhan: Central China Normal Univ Press, 2019; a draft is available at www.pdmi.ras.ru/~inp/ccNOTES.pdf
3	T H Cormen, C E Leiserson, R L Rivest, C. Stein Introduction to Algorithms. 3rd ed. Cambridge: The MIT Press, 2009
4	P A Ferguson, A. Turull Algebraic decomposition of commutative association schemes. J Algebra, 1985, 96: 211–229 https://doi.org/10.1016/0021-8693(85)90047-X
5	K Friedl, L. Ronyai Polynomial time solutions of some problems of computational algebra. In: Proc of the Seventeenth ACM STOC. 1985, 153–162 https://doi.org/10.1145/22145.22162
6	R Hammack, W Imrich, S. Klavzar Handbook of Product Graphs. Boca Raton: CRC Press, 2011 https://doi.org/10.1201/b10959
7	D G. Higman Coherent configurations. I. Rend Semin Mat Univ Padova, 1971, 44: 1–25
8	D G. Higman Coherent algebras. Linear Algebra Appl, 1987, 93: 209–239 https://doi.org/10.1016/S0024-3795(87)90326-0
9	N Kayal, T. Nezhmetdinov Factoring groups efficiently. Lecture Notes in Comput Sci, 2009, 5555: 585–596 https://doi.org/10.1007/978-3-642-02927-1_49
10	H. Krause Krull–Schmidt categories and projective covers. Expo Math, 2015, 33: 535–549 https://doi.org/10.1016/j.exmath.2015.10.001
11	C E Praeger, C. Schneider Permutation Groups and Cartesian Decompositions. Cambridge: Cambridge Univ Press, 2018 https://doi.org/10.1017/9781139194006
12	J B. Wilson Existence, algorithms, and asymptotics of direct product decompositions, I. Groups Complex Cryptol, 2012, 4(1): 1–39 https://doi.org/10.1515/gcc-2012-0007
13	B. Xu Direct products of association schemes and tensor products of table algebras. Algebra Colloq, 2013, 20(3): 475–494 https://doi.org/10.1142/S100538671300045X
14	P-H. Zieschang Theory of Association Schemes. Berlin: Springer, 2005

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