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Journal of Algebraic Combinatorics

, Volume 45, Issue 2, pp 525–552 | Cite as

Cartan coherent configurations

  • Ilia PonomarenkoEmail author
  • Andrey Vasil’ev
Article

Abstract

The Cartan scheme \(\mathcal{X}\) of a finite group G with a (BN)-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 right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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