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


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.


Vector Bundle Conjugacy Class Positive Root Dimension Vector Fundamental Region 
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© Institut des Hautes Études Scientifiques and Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK

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