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arXiv · 1602.07132

Cartan coherent configurations

Ilia Ponomarenko·Andrey Vasil'ev
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Abstract

The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is proved that if $G$ is a simple group of Lie type, then asymptotically, the coherent configuration $\cal X$ is 2-separable, i.e., the array of 2-dimensional intersection numbers determines $\cal X$ up to isomorphism. It is also proved that in this case, the base number of $\cal X$ equals 2. This enables us to construct a polynomial-time algorithm for recognizing the Cartan schemes when the rank of $G$ and order of the underlying field are sufficiently large. One of the key points in the proof of the main results is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.

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Ilia Ponomarenko, Andrey Vasil'ev. 2016-02-23. Cartan coherent configurations. https://doi.org/10.1007/s10801-016-0715-5

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