Skip to main content
SpringerLink
Log in

Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs

  • Research
  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections.

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

Similar content being viewed by others

References

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

  2. Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 4–6. Elements of Mathematics, Springer-Verlag, Berlin. Translated from the 1968 French original by Andrew Pressley (2002)

  3. Bugaenko, V., Cherniavsky, Y., Nagnibeda, T., Shwartz, R.: Weighted Coxeter graphs and generalized geometric representations of Coxeter groups. Discrete Appl. Math. 192, 17–27 (2015). https://doi.org/10.1016/j.dam.2014.05.012

    Article  MathSciNet  Google Scholar 

  4. Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. of Math. 35(2), 588–621 (1934). https://doi.org/10.2307/1968753

    Article  MathSciNet  Google Scholar 

  5. Coxeter, H.S.M.: The complete enumeration of finite groups of the form \(R_i^2=(R_iR_j)^{k_{ij}}=1\). J. London Math. Soc. 10, 21–25 (1935). https://doi.org/10.1112/jlms/s1-10.37.21

    Article  MathSciNet  Google Scholar 

  6. Dimitrov, I., Paquette, C., Wehlau, D., Xu, T.: Subregular J-rings of Coxeter systems via quiver path algebras. J. Algebra 612, 526–576 (2022). https://doi.org/10.1016/j.jalgebra.2022.09.003

  7. Donnelly, R.G.: Root systems for asymmetric geometric representations of Coxeter groups. Comm. Algebra 39, 1298–1314 (2011). https://doi.org/10.1080/00927871003662958

    Article  MathSciNet  Google Scholar 

  8. Hée, J.Y.: Système de racines sur un anneau commutatif totalement ordonné. Geom. Dedicata 37, 65–102 (1991). https://doi.org/10.1007/BF00150405

    Article  MathSciNet  Google Scholar 

  9. Hu, H.: Representations of Coxeter groups of Lusztig’s \(\varvec {a}\)-function value 1. Preprint. arXiv:2309.00593 (2023)

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

  11. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979). https://doi.org/10.1007/BF01390031

    Article  ADS  MathSciNet  Google Scholar 

  12. Krammer, D.: The conjugacy problem for Coxeter groups. Groups Geom. Dyn. 3, 71–171 (2009). https://doi.org/10.4171/GGD/52

    Article  MathSciNet  Google Scholar 

  13. Lusztig, G.: Some examples of square integrable representations of semisimple \(p\)-adic groups. Trans. Amer. Math. Soc. 277, 623–653 (1983). https://doi.org/10.2307/1999228

    Article  MathSciNet  Google Scholar 

  14. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park, CA (1984)

    Google Scholar 

  15. Serre, J.P.: Linear Representations of Finite Groups. volume 42 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg. Translated from the second French edition by Leonard L. Scott. (1977)

  16. Sunada, T.: Topological Crystallography. With a View Towards Discrete Geometric Analysis. volume 6 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, Tokyo (2013)

  17. Vinberg, E.B.: Discrete linear groups generated by reflections. Math. USSR, Izv. 5, 1083–1119 (1971). https://doi.org/10.1070/IM1971v005n05ABEH001203

Download references

Acknowledgements

The author is deeply grateful to Professor Nanhua Xi for his patient guidance and insightful discussions. The author would also like to thank Tao Gui for valuable exchanges.

Funding

The author did not receive support from any organization for the submitted work.

Author information

Authors and Affiliations

  1. Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing, 100871, China

    Hongsheng Hu

Authors
  1. Hongsheng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongsheng Hu.

Ethics declarations

Conflicts of interest

The author has no relevant interests to declare.

Additional information

Presented by: Andrew Mathas

Publisher's Note

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

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

Hu, H. Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs. Algebr Represent Theor 27, 961–994 (2024). https://doi.org/10.1007/s10468-023-10242-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-023-10242-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation