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Indecomposable parabolic bundles

and the Existence of Matrices in Prescribed Conjugacy Class Closures with Product Equal to the Identity

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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.

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References

  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. M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Am. Math. Soc.,85 (1957), 181–207.

    Google Scholar 

  3. P. Belkale, Local systems on P1-S for S a finite set, Compos. Math.,129 (2001), 67–86.

    Google Scholar 

  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. 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.

  6. D. Chan and C. Ingalls, Non-commutative coordinate rings and stacks, Proc. London Math. Soc.,88 (2004), 63–88.

  7. W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compos. Math.,126 (2001), 257–293.

    Google Scholar 

  8. W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann.,325 (2003), 55–79.

    Google Scholar 

  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. W. Crawley-Boevey and J. Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math.,553 (2002), 201–220.

    Google Scholar 

  11. P. Deligne, Equations différentielles à points singuliers réguliers, Lect. Notes Math.,163, Springer, Berlin (1970).

  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. 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. 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.

  15. M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices, III, Ann. Math.,70 (1959), 167–205.

    Google Scholar 

  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.

  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. V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math.,56 (1980), 57–92.

    Google Scholar 

  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.

  20. N. M. Katz, Rigid local systems, Princeton University Press, Princeton, NJ (1996).

  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. V. P. Kostov, On the Deligne-Simpson problem, C. R. Acad. Sci., Paris, Sér. I, Math.,329 (1999), 657–662.

  23. V. P. Kostov, On some aspects of the Deligne-Simpson problem, J. Dynam. Control Systems,9 (2003), 393–436.

    Google Scholar 

  24. V. P. Kostov, The Deligne-Simpson problem – a survey, preprint math.RA/0206298.

  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.

  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.

  27. B. Malgrange, Regular connections, after Deligne, chapter IV of A. Borel et al., Algebraic D-modules, Acad. Press, Boston (1987), 151–172.

  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. 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.

  30. A. Mihai, Sur le résidue et la monodromie d’une connexion méromorphe, C. R. Acad. Sci., Paris, Sér. A,281 (1975), 435–438.

  31. A. Mihai, Sur les connexions méromorphes, Rev. Roum. Math. Pures Appl.,23 (1978), 215–232.

    Google Scholar 

  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. C. M. Ringel, Tame algebras and integral quadratic forms, Lect. Notes Math.,1099, Springer, Berlin (1984).

  34. L. L. Scott, Matrices and cohomology, Ann. Math.,105 (1977), 473–492.

    Google Scholar 

  35. C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque,98 (1982), 1–209.

  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.

  37. K. Strambach and H. Völklein, On linearly rigid tuples, J. Reine Angew. Math.,510 (1999), 57–62.

    Google Scholar 

  38. H. Völklein, The braid group and linear rigidity, Geom. Dedicata,84 (2001), 135–150.

    Google Scholar 

  39. A. Weil, Generalization de fonctions abeliennes, J. Math. Pures Appl.,17 (1938), 47–87.

    Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    William Crawley-Boevey

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  1. William Crawley-Boevey
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Correspondence to William Crawley-Boevey.

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Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday

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Crawley-Boevey, W. Indecomposable parabolic bundles. Publ. Math., Inst. Hautes Étud. Sci. 100, 171–207 (2004). https://doi.org/10.1007/s10240-004-0025-7

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  • DOI: https://doi.org/10.1007/s10240-004-0025-7

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