Advertisement

Publications mathématiques de l'IHÉS

, Volume 100, Issue 1, pp 171–207 | Cite as

Indecomposable parabolic bundles

and the Existence of Matrices in Prescribed Conjugacy Class Closures with Product Equal to the Identity
  • William Crawley-Boevey
Article

Abstract

We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz’s middle convolution operation.

Keywords

Vector Bundle Conjugacy Class Positive Root Dimension Vector Fundamental Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5 (1998), 817–836.Google Scholar
  2. 2.
    M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Am. Math. Soc.,85 (1957), 181–207.Google Scholar
  3. 3.
    P. Belkale, Local systems on P1-S for S a finite set, Compos. Math.,129 (2001), 67–86.Google Scholar
  4. 4.
    I. Biswas, A criterion for the existence of a flat connection on a parabolic vector bundle, Adv. Geom.,2 (2002), 231–241.Google Scholar
  5. 5.
    S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, Representation Theory II (Ottawa, 1979), V. Dlab and P. Gabriel (eds.), Lect. Notes Math.,832, Springer, Berlin (1980), 103–169.Google Scholar
  6. 6.
    D. Chan and C. Ingalls, Non-commutative coordinate rings and stacks, Proc. London Math. Soc.,88 (2004), 63–88.Google Scholar
  7. 7.
    W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compos. Math.,126 (2001), 257–293.Google Scholar
  8. 8.
    W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann.,325 (2003), 55–79.Google Scholar
  9. 9.
    W. Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J.,118 (2003), 339–352.Google Scholar
  10. 10.
    W. Crawley-Boevey and J. Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math.,553 (2002), 201–220.Google Scholar
  11. 11.
    P. Deligne, Equations différentielles à points singuliers réguliers, Lect. Notes Math.,163, Springer, Berlin (1970).Google Scholar
  12. 12.
    M. Dettweiler and S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symb. Comput.,30 (2000), 761–798.Google Scholar
  13. 13.
    M. Furuta and B. Steer, Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math.,96 (1992), 38–102.Google Scholar
  14. 14.
    W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras, Singularities, representations of algebras, and vector bundles (Lambrecht, 1985), G.-M. Greuel and G. Trautmann (eds.), Lect. Notes Math.,1273, Springer, Berlin (1987), 265–297.Google Scholar
  15. 15.
    M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices, III, Ann. Math.,70 (1959), 167–205.Google Scholar
  16. 16.
    A. Haefliger, Local theory of meromorphic connections in dimension one (Fuchs theory), chapter III of A. Borel et al., Algebraic D-modules, Acad. Press, Boston (1987), 129–149.Google Scholar
  17. 17.
    D. Happel, I. Reiten and S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Am. Math. Soc.,120, no. 575 (1996).Google Scholar
  18. 18.
    V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math.,56 (1980), 57–92.Google Scholar
  19. 19.
    V. G. Kac, Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 1982), F. Gherardelli (ed.), Lect. Notes Math.,996, Springer, Berlin (1983), 74–108.Google Scholar
  20. 20.
    N. M. Katz, Rigid local systems, Princeton University Press, Princeton, NJ (1996).Google Scholar
  21. 21.
    V. P. Kostov, On the existence of monodromy groups of Fuchsian systems on Riemann’s sphere with unipotent generators, J. Dynam. Control Systems,2 (1996), 125–155.Google Scholar
  22. 22.
    V. P. Kostov, On the Deligne-Simpson problem, C. R. Acad. Sci., Paris, Sér. I, Math.,329 (1999), 657–662.Google Scholar
  23. 23.
    V. P. Kostov, On some aspects of the Deligne-Simpson problem, J. Dynam. Control Systems,9 (2003), 393–436.Google Scholar
  24. 24.
    V. P. Kostov, The Deligne-Simpson problem – a survey, preprint math.RA/0206298.Google Scholar
  25. 25.
    H. Kraft and Ch. Riedtmann, Geometry of representations of quivers, Representations of algebras (Durham, 1985), P. Webb (ed.) Lond. Math. Soc. Lect. Note Ser.,116, Cambridge Univ. Press (1986), 109–145.Google Scholar
  26. 26.
    H. Lenzing, Representations of finite dimensional algebras and singularity theory, Trends in ring theory (Miskolc, Hungary, 1996), Canadian Math. Soc. Conf. Proc.,22 (1998), Am. Math. Soc., Providence, RI (1998), 71–97.Google Scholar
  27. 27.
    B. Malgrange, Regular connections, after Deligne, chapter IV of A. Borel et al., Algebraic D-modules, Acad. Press, Boston (1987), 151–172.Google Scholar
  28. 28.
    V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structure, Math. Ann.,248 (1980), 205–239.Google Scholar
  29. 29.
    H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, Representations of algebras (Ottawa, 1992), Can. Math. Soc. Conf. Proc.,14 (1993), Am. Math. Soc., Providence, RI (1993), 313–337.Google Scholar
  30. 30.
    A. Mihai, Sur le résidue et la monodromie d’une connexion méromorphe, C. R. Acad. Sci., Paris, Sér. A,281 (1975), 435–438.Google Scholar
  31. 31.
    A. Mihai, Sur les connexions méromorphes, Rev. Roum. Math. Pures Appl.,23 (1978), 215–232.Google Scholar
  32. 32.
    O. Neto and F. C. Silva, Singular regular differential equations and eigenvalues of products of matrices, Linear Multilinear Algebra,46 (1999), 145–164.Google Scholar
  33. 33.
    C. M. Ringel, Tame algebras and integral quadratic forms, Lect. Notes Math.,1099, Springer, Berlin (1984).Google Scholar
  34. 34.
    L. L. Scott, Matrices and cohomology, Ann. Math.,105 (1977), 473–492.Google Scholar
  35. 35.
    C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque,98 (1982), 1–209.Google Scholar
  36. 36.
    C. T. Simpson, Products of Matrices, Differential geometry, global analysis, and topology (Halifax, NS, 1990), Can. Math. Soc. Conf. Proc.,12 (1992), Am. Math. Soc., Providence, RI (1991), 157–185.Google Scholar
  37. 37.
    K. Strambach and H. Völklein, On linearly rigid tuples, J. Reine Angew. Math.,510 (1999), 57–62.Google Scholar
  38. 38.
    H. Völklein, The braid group and linear rigidity, Geom. Dedicata,84 (2001), 135–150.Google Scholar
  39. 39.
    A. Weil, Generalization de fonctions abeliennes, J. Math. Pures Appl.,17 (1938), 47–87.Google Scholar

Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK

Personalised recommendations