Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity. (English) Zbl 1065.14040
Consider the Deligne-Simpson problem: For which tuples of conjugacy classes \(C_j\subset \text{GL}(n,{\mathbb C})\) do there exist irreducible tuples of matrices \(A_j\in C_j\) whose product is the identity matrix? The author considers a weaker version of the problem in which one requires the matrices \(A_j\) to belong only to the closures of the classes \(C_j\) and their tuple is not necessarily irreducible. To obtain the answer to this weaker problem he studies the possible dimension vectors of indecomposable parabolic bundles on the projective line. Both answers depend on the root system of a Kac-Moody Lie algebra.
The proofs use C. M. Ringel’s theory of tubular algebras [Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)], A. Mihai’s work on the existence of logarithmic connections [C. R. Acad. Sci., Paris, Sér. A 281, 435–438 (1975; Zbl 0323.53024) and Rev. Roum. Math. Pures Appl. 23, 215–232 (1978; Zbl 0379.53033)], the Riemann-Hilbert correspondence, and an algebraic version of Katz’s middle convolution operation [cf. M. Dettweiler and S. Reiter, J. Symb. Comput. 30, 761–798 (2000; Zbl 1049.12005)]. See also the survey on the Deligne-Simpson problem given by the reviewer [J. Algebra 281, 83–108 (2004; Zbl 1066.15016)].
The proofs use C. M. Ringel’s theory of tubular algebras [Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)], A. Mihai’s work on the existence of logarithmic connections [C. R. Acad. Sci., Paris, Sér. A 281, 435–438 (1975; Zbl 0323.53024) and Rev. Roum. Math. Pures Appl. 23, 215–232 (1978; Zbl 0379.53033)], the Riemann-Hilbert correspondence, and an algebraic version of Katz’s middle convolution operation [cf. M. Dettweiler and S. Reiter, J. Symb. Comput. 30, 761–798 (2000; Zbl 1049.12005)]. See also the survey on the Deligne-Simpson problem given by the reviewer [J. Algebra 281, 83–108 (2004; Zbl 1066.15016)].
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 15A24 | Matrix equations and identities |
| 16G10 | Representations of associative Artinian rings |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
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