Skip to main content
SpringerLink
Log in

Monodromy of Fuchsian systems on complex linear spaces

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.

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. A.-M. Aubert and P. Achar, “On Rank-Two Complex Reflection Groups,” Prépubl. No. 383 (Inst. Math. Jussieu, 2004), http://www.institut.math.jussieu.fr/:_preprints/pdf/383.pdf

  2. N. Bourbaki, Éléments de mathématique, Fasc. 34: Groupes et algèbres de Lie (Hermann, Paris, 1968; Mir, Moscow, 1972), Chs. 4–6.

    Google Scholar 

  3. M. Broué, G. Malle, and R. Rouquier, “Complex Reflection Groups, Braid Groupes, Hecke Algebras,” J. Reine Angew. Math. 500, 127–190 (1998).

    MATH  MathSciNet  Google Scholar 

  4. N. D. Vasilevich and N. N. Ladis, “The Integrability of Pfaffian Equations on ℂPn,” Diff. Uravn. 15(4), 732–733 (1979) [Diff. Eqns. 15, 513–514 (1979)].

    MathSciNet  Google Scholar 

  5. I. V. Cherednik, “Generalized Braid Groups and Local r-Matrix Systems,” Dokl. Akad. Nauk SSSR 307(1), 49–53 (1989) [Sov. Math. Dokl. 40 (1), 43–48 (1990)].

    MathSciNet  Google Scholar 

  6. I. Cherednik, “Monodromy Representations for Generalized Knizhnik-Zamolodchikov Equations and Hecke Algebras,” Publ. RIMS, Kyoto Univ. 27(5), 711–726 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. M. Cohen, “Finite Complex Reflection Groups,” Ann. Sci. Ecole Norm. Super., Sér. 4, 9, 379–436 (1976).

    MATH  Google Scholar 

  8. H. S. M. Coxeter, “The Symmetry Groups of the Regular Complex Polygons,” Arch. Math. 13, 86–97 (1962).

    Article  MathSciNet  Google Scholar 

  9. P. Etingof and E. Rains, “Central Extensions of Preprojective Algebras, the Quantum Heisenberg Algebra, and 2-Dimensional Complex Reflection Groups,” math.RT/0503393.

  10. G. Giorgadze, “Monodromy Approach to Quantum Computing,” Int. J. Mod. Phys. B 16(30), 4593–4605 (2002).

    Article  MATH  Google Scholar 

  11. A. B. Givental’, “Twisted Picard-Lefschetz Formulas,” Funkts. Anal. Prilozh. 22(1), 12–22 (1988) [Funct. Anal. Appl. 22 (1), 10–18 (1988)].

    MathSciNet  Google Scholar 

  12. M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, “Topological Quantum Computation,” quant-ph/0101025.

  13. M. Hughes, “Extended Root Graphs for Complex Reflection Groups,” Commun. Algebra 27, 119–148 (1999).

    MATH  Google Scholar 

  14. M. C. Hughes and A. O. Morris, “Root Systems for Two Dimensional Complex Reflection Groups,” Sèmin. Lothar. Comb. 45, B45e (2001).

    Google Scholar 

  15. T. Kohno, “Linear Representations of Braid Groups and Classical Yang-Baxter Equations,” Contemp. Math. 78, 339–363 (1988).

    MathSciNet  Google Scholar 

  16. V. P. Lexin, “WDVV Equations and Generalized Burau Representations,” in Mathematical Ideas of P.L. Chebyshev and Their Application to Modern Problems of Natural Science, Obninsk, May 14–18, 2002: Abstracts of Papers of the International Conference, pp. 59–60.

  17. V. P. Lexin, “Monodromy of Rational KZ Equations and Some Related Topics,” Acta Appl. Math. 75, 105–115 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  18. V. P. Leksin, “Monodromy of Cherednik-Kohno-Veselov Connections,” Proc. IRMA (2006) (in press).

  19. G. Nebe, “The Root Lattices of the Complex Reflection Groups,” J. Group Theory 2, 15–38 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  20. E. M. Opdam, Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups (Math. Soc. Japan, Tokyo, 2000), MSJ Memoirs 8.

    MATH  Google Scholar 

  21. G. C. Shephard and J. A. Todd, “Finite Unitary Reflection Groups,” Can. J. Math. 6, 274–304 (1954).

    MATH  MathSciNet  Google Scholar 

  22. C. C. Squier, “Matrix Representations of Artin Groups,” Proc. Am. Math. Soc. 103, 49–53 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  23. V. Toledano Laredo, “Flat Connections and Quantum Groups,” Acta Appl. Math. 73, 155–173 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  24. A. P. Veselov, “On Geometry of a Special Class of Solutions to Generalized WDVV Equations,” in Integrability: The Seiberg-Witten and Whitham Equations, Edinburg, 1998 (Gordon and Breach, Amsterdam, 2000), pp. 125–135.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Algebra and Geometry, Kolomna State Pedagogical Institute, ul. Zelenaya 30, Kolomna, 140411, Russia

    V. P. Leksin

Authors
  1. V. P. Leksin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.P. Leksin, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 256, pp. 267–277.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leksin, V.P. Monodromy of Fuchsian systems on complex linear spaces. Proc. Steklov Inst. Math. 256, 253–262 (2007). https://doi.org/10.1134/S0081543807010142

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543807010142

Keywords

Search

Navigation