Twisted Weak Orders of Coxeter Groups


In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such orders and the infinite reduced words. We show that for a 2 closure biclosed set B in Φ+, the B-twisted weak order is a non-complete meet semilattice if B is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups.

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


  1. 1.

    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Volume 231 of Graduate Texts in Mathematics. Springer, Berlin (2005)

    Google Scholar 

  2. 2.

    Dyer, M.J.: Quotients of twisted bruhat orders. J. Algebra 163, 861–879 (1994)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dyer, M.J.: Hecke algebras and shellings of bruhat intervals. II. twisted bruhat orders. Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989) 141–165, Contemp Math. 139, Amer. Math. Soc., Providence (1992)

  4. 4.

    Dyer, M., Lehrer, G.: Reflection subgroups of finite and affine weyl groups. Trans. Am. Math. Soc. 363(11), 5971–6005 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dyer, M.: On the weak order of Coxeter groups. arXiv:abs/1108.5557. To appear in Canadian Journal of Mathematics (2011)

  6. 6.

    Dyer, M.: Reflection orders of affine weyl groups (2017)

  7. 7.

    Doković, D.Z.̌, Check, P., Hée, J.-Y.: On closed subsets of root systems. Canad. Math. Bull. 37(3), 338–345 (1994)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Edgar, T.: Sets of reflections defining twisted Bruhat orders. J. Algebr. Comb. 26, 357–362 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gobet, T.: Twisted filtrations of Soergel bimodules and linear Rouquier complexes. J. Algebra 484, 275–309 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hohlweg, C., Labbé, J.-P.: On inversion sets and the weak order in Coxeter groups. European J. Combin. 55, 1–19 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

  12. 12.

    Lam, T., Thomas, A.: Infinite reduced words and the Tits boundary of a Coxeter group. Int. Math. Res. Not. 17, 7690–7733 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Pilkington, A.: Convex geometries on root systems. Comm. Algebra 34(9), 3183–3202 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wang, W.: Infinite reduced words, lattice property and braid graph of affine Weyl groups. arXiv:1803.03017 (2018)

Download references


The author acknowledges the support from Guangdong Natural Science Foundation Project 2018A030313581. Some results of the paper are based on part of the author’s thesis. The author wishes to thank his advisor Matthew Dyer, who was aware of the twisted weak order while studying the twisted Bruhat order, for his guidance. The author thanks the anonymous referees for their time and useful comments for improving the paper.

Author information



Corresponding author

Correspondence to Weijia Wang.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, W. Twisted Weak Orders of Coxeter Groups. Order 36, 511–523 (2019).

Download citation


  • Coxeter groups
  • Twisted weak order
  • Closure operator
  • Lattice