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On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries

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Abstract

Given a (thick) irreducible spherical building \(\Omega \), we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type \({{\textsf{A}}}_n\), and the case of type \(\mathsf {F_{4,4}}\) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).

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Acknowledgements

We are grateful to Arjeh Cohen for helping to clarify the state of the art, to Antonio Pasini for many helpful remarks on an earlier version of the manuscript, and to You Qi for a discussion on adjoint representations and the Weyl module. Also, part of this research was done while the first author was a Leibniz Fellow at the Mathematisches Forschungsinstitut Oberwolfach in 2021, gratefully enjoying the facilities and stimulating environment of the institute.

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Authors and Affiliations

  1. Department of Mathematics, Ghent University, Krijgslaan 281–S25, 9000, Ghent, Belgium

    A. De Schepper & H. Van Maldeghem

  2. Department of Mathematics, University of Auckland, 38 Princes Street, 1010, Auckland, New Zealand

    J. Schillewaert

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  1. A. De Schepper
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  2. J. Schillewaert
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  3. H. Van Maldeghem
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Correspondence to J. Schillewaert.

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The first author is supported by the Fund for Scientific Research Flanders—FWO Vlaanderen 12ZJ220N

The second author is supported by the New Zealand Marsden fund grant MFP-UOA2122.

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De Schepper, A., Schillewaert, J. & Van Maldeghem, H. On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries. Combinatorica 44, 355–392 (2024). https://doi.org/10.1007/s00493-023-00075-y

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  • DOI: https://doi.org/10.1007/s00493-023-00075-y

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