Skip to main content
SpringerLink
Log in

b-Functions associated with quivers of type A

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.

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. S. Abeasis, Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type A n , Trans. Amer. Math. Soc. 282 (1984), 463–485.

    MathSciNet  MATH  Google Scholar 

  2. S. Abeasis, A. Del Fra, Degenerations for the representations of a quiver of type \(\mathcal{A}_m\), J. Algebra 93 (1985), 376–412.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Brion, Representations of quivers, in: Geometric Methods in Representation Theory, Grenoble, 2008, Lecture Notes of the Summer School, available online.

  4. A. S. Buch, R. Rimányi, A formula for nonequioriented quiver orbits of type A, J. Algebraic Geom. 16 (2007), 531–546.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Derksen, J. Weyman, Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients, J. of the AMS 13 (2000), 467–479.

    MathSciNet  MATH  Google Scholar 

  6. H. Derksen, J. Weyman, Quiver representations, Notices of the AMS 52 (2005), no. 2, 200–206.

    MathSciNet  MATH  Google Scholar 

  7. A. Knutson, E. Miller, M. Shimozono, Four positive formulae for type A quiver polynomials, Invent. Math. 166 (2006), 229–325.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Granger, M. Schulze, On the symmetry of b-functions of linear free divisors, Publ. RIMS 46 (2010), 479–506.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Gyoja, Theory of prehomogeneous vector spaces without regularity condition, Publ. RIMS 27 (1991), 861–922.

    Article  MathSciNet  MATH  Google Scholar 

  10. V. G. Kac, Root systems, representations of quivers and invariant theory, in: Invariant Theory, Montecatini, 1982, Lecture Notes in Mathematics, Vol. 996, Springer, Berlin, 1983, pp. 74–108.

    Chapter  Google Scholar 

  11. M. Kashiwara, D-modules and Microlocal Calculus, Translations of Mathematical Monographs, Vol. 217, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  12. T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical Monographs, Vol. 215, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  13. K. Koike, Relative invariants of the polynomial rings over Type A r , Ã r quivers, Adv. Math. 86 (1991), 235–262.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Mortajine, Classiffication des Espaces Préhomogènes de Type Parabolique Réguliers et de leurs Invariants Relatifs, Travaux en Cours 40 (1991), Hermann, Paris.

  15. C. M. Ringel, Representations of K-species and bimodules, J. Algebra 41 (1976), 269–302.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Rubenthaler, Algèbres de Lie et Espaces Préhomogènes, Travaux en Cours 44 (1992), Hermann, Paris.

  17. F. Sato, K. Sugiyama, Multiplicity one property and the decomposition of b-functions, Internat. J. Math. 17 (2006), 195–229.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Sato, Theory of prehomogeneous vector spaces (algebraic part), notes by Takuro Shintani, translated from the Japanese by Masakazu Muro, Nagoya Math. J. 120 (1990), 1–34.

    MathSciNet  MATH  Google Scholar 

  19. A. Schofield, Semi-invariants of quivers, J. London Math. Soc. 43 (1991), 385–395.

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Sevenheck, Bernstein polynomials and spectral numbers for linear free divisors, preprint, arXiv:0905.0971.

  21. D. A. Shmelkin, Locally semisimple representations of quivers, Transform. Groups 12 (2007), 153–173.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. A. Shmelkin, Some algorithms for semi-invariants of quivers, J. Algebra 322 (2009), 3997–4010.

    Article  MathSciNet  MATH  Google Scholar 

  23. D. A. Shmelkin, TETIVA, a computer program available at http://www.mccme.ru/~mitia

  24. K. Sugiyama, b-Function of a prehomogeneous vector space with no regular component, Comment. Math. Univ. St. Pauli 54 (2005), 99–119.

    MathSciNet  MATH  Google Scholar 

  25. K. Ukai, b-Functions of prehomogeneous vector spaces of Dynkin–Kostant type for exceptional groups, Compositio Math. 135 (2003), 49–101.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Wachi, Logarithmic derivative and the Capelli identities, preprint, 2009.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chiba Institute of Technology, 2–1–1, Shibazono, Narashino Chiba, 275–0023, Japan

    Kazunari Sugiyama

Authors
  1. Kazunari Sugiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazunari Sugiyama.

Additional information

Dedicated to the 60th birthday of Professor Tatsuo Kimura

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sugiyama, K. b-Functions associated with quivers of type A. Transformation Groups 16, 1183–1222 (2011). https://doi.org/10.1007/s00031-011-9135-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-011-9135-8

Keywords

Search

Navigation