Skip to main content
SpringerLink
Log in

On automorphisms of undirected Bruhat graphs

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

The undirected Bruhat graph \(\Gamma (u,v)\) has the elements of the Bruhat interval [u, v] as vertices, with edges given by multiplication by a reflection. Famously, \(\Gamma (e,v)\) is regular if and only if the Schubert variety \(X_v\) is smooth, and this condition on v is characterized by pattern avoidance. In this work, we classify when \(\Gamma (e,v)\) is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of \(\Gamma (u,v)\) in the course of which we show that special matchings, which originally appeared in the theory of Kazhdan–Lusztig polynomials, can be characterized, for the symmetric and right-angled groups, as certain \(\Gamma (u,v)\)-automorphisms which are conjecturally sufficient to generate the orbit of e under \({{\,\textrm{Aut}\,}}(\Gamma (e,v))\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability

The work in this publication does not rely on any data.

References

  1. Abdallah, N., Hansson, M., Hultman, A.: Topology of posets with special partial matchings. Adv. Math. 348, 255–276 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Billey, S., Postnikov, A.: Smoothness of Schubert varieties via patterns in root subsystems. Adv. Appl. Math. 34(3), 447–466 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate texts in mathematics, vol. 231. Springer, New York (2005)

    MATH  Google Scholar 

  4. Brenti, F.: A combinatorial formula for Kazhdan-Lusztig polynomials. Invent. Math. 118(2), 371–394 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenti, F.: The intersection cohomology of Schubert varieties is a combinatorial invariant. Eur. J. Combin. 25(8), 1151–1167 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenti, F., Caselli, F., Marietti, M.: Special matchings and Kazhdan-Lusztig polynomials. Adv. Math. 202(2), 555–601 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caselli, F., Marietti, M.: Special matchings in Coxeter groups. Eur. J. Combin. 61, 151–166 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caselli, F., Marietti, M.: A simple characterization of special matchings in lower Bruhat intervals. Discr. Math. 341(3), 851–862 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charles, B., Lars, B., Alex, D., Petar, V., Geordie, W.: Towards combinatorial invariance for Kazhdan-Lusztig polynomials. arXiv preprint arXiv:2111.15161, (2021)

  10. Christian, G., Yibo, G.: Self-dual intervals in the Bruhat order. Selecta Math. (N.S.), 26(5):Paper No. 77, 23, (2020)

  11. Davies, A., Veličković, P., Buesing, L., Blackwell, S., Zheng, D., Tomašev, N., Tanburn, R., Battaglia, P., Blundell, C., Juhász, A., et al.: Advancing mathematics by guiding human intuition with ai. Nature 600(7887), 70–74 (2021)

    Article  MATH  Google Scholar 

  12. Dyer, M.: On the “Bruhat graph’’ of a Coxeter system. Compos. Math. 78(2), 185–191 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Dyer, M.J.: Hecke algebras and shellings of Bruhat intervals. Compos. Math. 89(1), 91–115 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Francesco, B., Fabrizio, C., Mario, M.: Diamonds and Hecke algebra representations. Int. Math. Res. Not., pages Art. ID 29407, 34, (2006)

  15. Gasharov, V.: Factoring the Poincaré polynomials for the Bruhat order on \(S_n\). J. Combin. Theory Ser. A 83(1), 159–164 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guillemin, V., Holm, T., Zara, C.: A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebr. Combin. 23(1), 21–41 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. James, B.: Carrell. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. In Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), volume 56 of Proc. Sympos. Pure Math., pages 53–61. Amer. Math. Soc., Providence, RI, (1994)

  19. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lakshmibai, V., Sandhya, B.: Criterion for smoothness of Schubert varieties in \({\rm Sl}(n)/B\). Proc. Indian Acad. Sci. Math. Sci. 100(1), 45–52 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marietti, M.: Algebraic and combinatorial properties of zircons. J. Algebraic Combin. 26(3), 363–382 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Richmond, E., Slofstra, W.: Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties. Math. Ann. 366(1–2), 31–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Suho, O., Postnikov, A., Yoo, H.: Bruhat order, smooth Schubert varieties, and hyperplane arrangements. J. Combin. Theory Ser. A 115(7), 1156–1166 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tits, J.: Le problème des mots dans les groupes de Coxeter. In: Symposia Mathematica (INDAM. Rome, 1967/68), vol. 1, pp. 175–185. Academic Press, London (1969)

  25. Waterhouse, W.C.: Automorphisms of the Bruhat order on Coxeter groups. Bull. Lond. Math. Soc. 21(3), 243–248 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are very grateful to Thomas Lam and Grant Barkley for their helpful comments and suggestions. We also wish to thank Mario Marietti for alerting us to important references.

Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Christian Gaetz

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Yibo Gao

Authors
  1. Christian Gaetz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yibo Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Gaetz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

C.G. was supported by a Klarman Postdoctoral Fellowship at Cornell University and by a National Science Foundation Postdoctoral Research Fellowship (DMS-2103121).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gaetz, C., Gao, Y. On automorphisms of undirected Bruhat graphs. Math. Z. 303, 31 (2023). https://doi.org/10.1007/s00209-022-03194-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00209-022-03194-2

Search

Navigation