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Cartan coherent configurations

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Abstract

The Cartan scheme \(\mathcal{X}\) 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 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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Fig. 1
Fig. 2

Notes

  1. In the complete colored graph representing \(\mathcal{X}\), k is the maximum number of the monochrome arcs incident to a vertex, and c is the maximum number of triangles with fixed base; the other two sides of which are monochrome arcs.

  2. In the case of Suzuki and Ree groups, \(q=2^{2\alpha +1}\) for \({}^2B_2(q)\) and \({}^2F_4(q)\), and \(q=3^{2\alpha +1}\) for \({}^2G_2(q)\), where \(\alpha >1\) is an integer.

  3. The alternative way to establish the same is to apply Zenkov’s theorem [22]. It yields that since H is abelian, there is an element \(g_0\in G\) such that \(H\cap H^{g_0}\) lies in the Fitting subgroup of G, which is trivial if the group G is simple.

  4. It is worth noting that in ‘an asymptotical sense’ the required lower bounds can be taken from [15, Lemma 3.4]. We choose to use the results of the later paper [5] in order to obtain the numerical values of \(l_0\) and a.

  5. It is worth mentioning that despite [5, Tables 3.7–3.9] contain the bounds on the sizes of conjugacy classes in the group \({\text {Inndiag}}(G)\) rather than G itself, the bounds for \(m_0\) in Table 5 are correct, because \(|G:C_G(h)|=|{\text {Inndiag}}(G):C_{{\text {Inndiag}}(G)}(h)|\) for every \(h\in H\) (see, e.g., the definition of the diagonal automorphism in [6, Sec. 12.2]).

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Authors and Affiliations

  1. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    Ilia Ponomarenko

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Andrey Vasil’ev

  3. Novosibirsk State University, Novosibirsk, Russia

    Andrey Vasil’ev

Authors
  1. Ilia Ponomarenko
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  2. Andrey Vasil’ev
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Correspondence to Ilia Ponomarenko.

Additional information

The work of the first and the second authors was partially supported, respectively, by the Grant RFBR No. 14-01-00156 and RFFI Grant No. 13-01-00505.

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Ponomarenko, I., Vasil’ev, A. Cartan coherent configurations. J Algebr Comb 45, 525–552 (2017). https://doi.org/10.1007/s10801-016-0715-5

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