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Proper elements of Coxeter groups

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Abstract

We extend the notion of proper elements to all finite Coxeter groups. For all infinite families of finite Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This proves a conjecture of the third author and Alexander Yong regarding the proportion of Schubert varieties that are Levi spherical for all infinite families of Weyl groups. We also enumerate the proper elements in the exceptional Coxeter groups.

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Acknowledgements

We thank Alex Yong for helpful discussions. We thank the anonymous referees for their helpful remarks on the organization of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    József Balogh

  2. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    David Brewster

  3. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Reuven Hodges

Authors
  1. József Balogh
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  2. David Brewster
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  3. Reuven Hodges
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All authors have contributed equally.

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Correspondence to Reuven Hodges.

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The project was completed as part of the ICLUE (Illinois Combinatorics Lab for Undergraduate Experience) program, which was funded by the NSF RTG Grant DMS 1937241. Research was partially supported by NSF Grant DMS 1764123, NSF RTG Grant DMS 1937241, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 18132), the Langan Scholar Fund (UIUC), and the Simons Fellowship. Part of this work was completed while RH was a postdoc at the University of Illinois at Urbana-Champaign. RH was partially supported by an AMS Simons Travel grant.

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Balogh, J., Brewster, D. & Hodges, R. Proper elements of Coxeter groups. European Journal of Mathematics 10, 32 (2024). https://doi.org/10.1007/s40879-024-00746-0

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  • DOI: https://doi.org/10.1007/s40879-024-00746-0

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