Local finiteness of the twisted Bruhat orders on affine Weyl groups

Weijia Wang 1
  • 1 School of Mathematics (Zhuhai), Sun Yat-sen University, Guangdong, Zhuhai, P. R. China
Weijia Wang
  • Corresponding author
  • School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong, P. R. China
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra163 (1994), 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set 𝐵 in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed 𝐵-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

  • [1]

    A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math. 231, Springer, New York, 2005.

  • [2]

    A. Björner, P. H. Edelman and G. M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), no. 3, 263–288.

    • Crossref
    • Export Citation
  • [3]

    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, 2nd ed., Encyclopedia Math. Appl. 46, Cambridge University, Cambridge, 1999.

  • [4]

    N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elem. Math. (Berlin), Springer, Berlin, 2002.

  • [5]

    J. R. Büchi and W. E. Fenton, Large convex sets in oriented matroids, J. Combin. Theory Ser. B 45 (1988), no. 3, 293–304.

    • Crossref
    • Export Citation
  • [6]

    Y. Chen and M. Dyer, On the combinatorics of B × B B\times B-orbits on group compactifications, J. Algebra 263 (2003), no. 2, 278–293.

    • Crossref
    • Export Citation
  • [7]

    D. Ž. Doković, P. Check and J.-Y. Hée, On closed subsets of root systems, Canad. Math. Bull. 37 (1994), no. 3, 338–345.

    • Crossref
    • Export Citation
  • [8]

    M. Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57–73.

    • Crossref
    • Export Citation
  • [9]

    M. Dyer, Reflection orders of affine Weyl groups, preprint (2014).

  • [10]

    M. Dyer, On the weak order of Coxeter groups, Canad. J. Math. 71 (2019), no. 2, 299–336.

    • Crossref
    • Export Citation
  • [11]

    M. J. Dyer, Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders, Kazhdan–Lusztig Theory and Related Topics (Chicago 1989), Contemp. Math. 139, American Mathematical Society, Providence (1992), 141–165.

  • [12]

    M. J. Dyer, Bruhat intervals, polyhedral cones and Kazhdan–Lusztig–Stanley polynomials, Math. Z. 215 (1994), no. 2, 223–236.

    • Crossref
    • Export Citation
  • [13]

    M. J. Dyer, Quotients of twisted Bruhat orders, J. Algebra 163 (1994), no. 3, 861–879.

    • Crossref
    • Export Citation
  • [14]

    M. J. Dyer and G. I. Lehrer, Reflection subgroups of finite and affine Weyl groups, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5971–6005.

    • Crossref
    • Export Citation
  • [15]

    T. Edgar, Universal reflection subgroups and exponential growth in Coxeter groups, Comm. Algebra 41 (2013), no. 4, 1558–1569.

    • Crossref
    • Export Citation
  • [16]

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

    • Crossref
    • Export Citation
  • [17]

    A. Hultman, The finite antichain property in Coxeter groups, Ark. Mat. 45 (2007), no. 1, 61–69.

    • Crossref
    • Export Citation
  • [18]

    J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.

  • [19]

    J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University, Cambridge, 1990.

  • [20]

    V. G. Kac, Infinite-dimensional Lie Algebras, 3rd ed., Cambridge University, Cambridge, 1990.

  • [21]

    D. E. Speyer, Powers of Coxeter elements in infinite groups are reduced, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1295–1302.

  • [22]

    W. Wang, Infinite reduced words, lattice property and braid graph of affine Weyl groups, J. Algebra 536 (2019), 170–214.

    • Crossref
    • Export Citation
  • [23]

    W. Wang, Twisted weak orders of Coxeter groups, Order 36 (2019), no. 3, 511–523.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

Search