Skip to main content
SpringerLink
Log in

The Steinberg torus of a Weyl group as a module over the Coxeter complex

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

Associated with each irreducible crystallographic root system \(\varPhi \), there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of \(\varPhi \). This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of \(\varPhi \). The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space spanned by affine descent classes of a Weyl group, over the space spanned by ordinary descent classes. The latter constitute a subalgebra of the group algebra, the classical descent algebra of Solomon. We provide combinatorial models for the module of faces when \(\varPhi \) is of type A or C.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Abramenko, P., Brown, K.S.: Buildings, Theory and Applications. Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)

  2. Aguiar, M., Bergeron, N., Nyman, K.: The peak algebra and the descent algebras of types \(B\) and \(D\). Trans. Am. Math. Soc. 356(7), 2781–2824 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Atkinson, M.D., Pfeiffer, G., Van Willigenburg, S.J.: The \(p\)-modular descent algebras. Algebras Represent. Theory 5(1), 101–113 (2002)

    Article  MATH  Google Scholar 

  4. Baumann, P., Hohlweg, C.: A Solomon descent theory for the wreath products \(G\wr \mathfrak{S}_n\). Trans. Am. Math. Soc. 360(3), 1475–1538 (2008). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bergeron, F., Bergeron, N., Howlett, R.B., Taylor, D.E.: A decomposition of the descent algebra of a finite Coxeter group. J. Algebr. Comb. 1(1), 23–44 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bidigare, T.P.: Hyperplane arrangement face algebras and their associated Markov chains. PhD thesis, University of Michigan (1997)

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

    Google Scholar 

  8. Bonnafé, C., Pfeiffer, G.: Around Solomon’s descent algebras. Algebras Represent. Theory 11(6), 577–602 (2008)

    Article  MATH  Google Scholar 

  9. Brown, K.S.: Semigroups, rings, and Markov chains. J. Theor. Probab. 13(3), 871–938 (2000)

    Article  MATH  Google Scholar 

  10. Brown, K.S., Diaconis, P.: Random walks and hyperplane arrangements. Ann. Probab. 26(4), 1813–1854 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cellini, P.: A general commutative descent algebra. J. Algebra 175(3), 990–1014 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dilks, K., Petersen, T.K., Stembridge, J.R.: Affine descents and the Steinberg torus. Adv. Appl. Math. 42(4), 423–444 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fulman, J.: Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting. J. Algebra 231(2), 614–639 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fulman, J.: Descent algebras, hyperplane arrangements, and shuffling cards. Proc. Am. Math. Soc. 129(4), 965–973 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garsia, A.M., Reutenauer, C.: A decomposition of Solomon’s descent algebra. Adv. Math. 77(2), 189–262 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Humphreys, J.E.: Reflection Groups and Coxeter Groups, Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  17. Lam, T., Postnikov, A.: Alcoved polytopes—I. Discrete Comput. Geom. 38(3), 453–478 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lam, T., Postnikov, A.: Alcoved polytopes—II. arXiv:1202.4015

  19. Mathas, A., Orellana, R.C.: Cyclotomic Solomon algebras. Adv. Math. 219(2), 450–487 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moszkowski, P.: Généralisation d’une formule de Solomon relative à l’anneau de groupe d’un groupe de Coxeter. CR Acad. Sci. Paris Sér. I Math. 309(8), 539–541 (1989)

    MATH  MathSciNet  Google Scholar 

  21. Patras, F.: L’algèbre des descentes d’une bigèbre graduée. J. Algebra 170(2), 547–566 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Petersen, T.K.: Cyclic descents and \(P\)-partitions. J. Algebr. Comb. 22(3), 343–375 (2005)

    Article  MATH  Google Scholar 

  23. Saliola, F.V.: On the quiver of the descent algebra. J. Algebra 320(11), 3866–3894 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  25. Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence, RI (1968)

  26. Tits, J.: Two properties of Coxeter complexes. J. Algebra 41(2), 265–268 (1976). Appendix to “A Mackey formula in the group ring of a Coxeter group” (J. Algebra 41 (1976), no. 2, 255–264) by Louis Solomon

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Marcelo Aguiar

  2. Department of Mathematical Sciences, DePaul University, Chicago, IL, 60614, USA

    T. Kyle Petersen

Authors
  1. Marcelo Aguiar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kyle Petersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Kyle Petersen.

Additional information

Aguiar supported in part by NSF Grant DMS-1001935.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aguiar, M., Petersen, T.K. The Steinberg torus of a Weyl group as a module over the Coxeter complex. J Algebr Comb 42, 1135–1175 (2015). https://doi.org/10.1007/s10801-015-0620-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-015-0620-3

Keywords

Mathematics Subject Classification

Search

Navigation