Indecomposable parabolic bundles
Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 171-207.

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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     author = {Crawley-Boevey, William},
     title = {Indecomposable parabolic bundles},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {171--207},
     publisher = {Springer},
     volume = {100},
     year = {2004},
     doi = {10.1007/s10240-004-0025-7},
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     zbl = {1065.14040},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-004-0025-7/}
}
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Crawley-Boevey, William. Indecomposable parabolic bundles. Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 171-207. doi : 10.1007/s10240-004-0025-7. http://archive.numdam.org/articles/10.1007/s10240-004-0025-7/

1. S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5 (1998), 817-836. | MR | Zbl

2. M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Am. Math. Soc., 85 (1957), 181-207. | MR | Zbl

3. P. Belkale, Local systems on P 1-S for S a finite set, Compos. Math., 129 (2001), 67-86. | MR | Zbl

4. I. Biswas, A criterion for the existence of a flat connection on a parabolic vector bundle, Adv. Geom., 2 (2002), 231-241. | MR | Zbl

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. | MR | Zbl

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

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

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

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. | MR | Zbl

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

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

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. | MR | Zbl

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. | MR | Zbl

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. | MR | Zbl

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

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). | MR | Zbl

18. V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math., 56 (1980), 57-92. | MR | Zbl

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. | MR | Zbl

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

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. | Zbl

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

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

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

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. | MR | Zbl

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. | MR | Zbl

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. | MR | Zbl

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. | MR | Zbl

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

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

32. O. Neto and F. C. Silva, Singular regular differential equations and eigenvalues of products of matrices, Linear Multilinear Algebra, 46 (1999), 145-164. | MR | Zbl

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

34. L. L. Scott, Matrices and cohomology, Ann. Math., 105 (1977), 473-492. | MR | Zbl

35. C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque, 98 (1982), 1-209. | Numdam | MR | Zbl

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. | MR | Zbl

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

38. H. Völklein, The braid group and linear rigidity, Geom. Dedicata, 84 (2001), 135-150. | MR | Zbl

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

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