Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity

Details Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity

2003-07-17

arxiv.org/abs/math/0307246v2

Mathematics

Algebraic Geometry

30 pages, various corrections and improvements

Scientific paper

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 by 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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