On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero

By W. Crawley‐Boevey. 2001

133 citations recorded by the source corpus.

We determine those $k$-tuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/85b1bef13e7e7dffb2096bec7308187139236b16

How researchers see this paper

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In Algebra and Number Theory

1. Vladimir Petrov Kostov
2. Daisuke Yamakawa
3. W. Crawley‐Boevey
4. V. Ginzburg
5. Carlos Simpson
6. P. Shaw
7. I. Losev
8. Bernard Leclerc

In Discrete Mathematics and Combinatorics

1. Daisuke Yamakawa
2. Vladimir Petrov Kostov
3. W. Crawley‐Boevey
4. Yuya Mizuno
5. Markus Reineke
6. Bernard Leclerc
7. Tamás Hausel
8. Osamu Iyama

In Geometry and Topology

1. Vladimir Petrov Kostov
2. Daisuke Yamakawa
3. W. Crawley‐Boevey
4. P. Boalch
5. I. Losev
6. H. Nakajima
7. V. Ginzburg
8. Bernard Leclerc

Citations in the corpus

• The Deligne–Simpson problem—a survey (2002, methodology): “T can be deduced from Theorem 28 from [41]; it is proved for matrices Aj in [12] as well.”
• Katz's middle convolution algorithm (2006, unknown): “, [63] [64] [45] [46] [55] [33].”
• Geometry of Multiplicative Preprojective Algebra (2007, methodology): “He solved the additive version [9] using this idea, and in [11], he and Shaw observed that the “multiplicative quiver variety” Φ−1 V (q)//GV describes the multiplicative analogue of the above variety : R := { (A1,A2, .”
• Indecomposable parabolic bundles (2004, methodology): “We have already solved the corresponding additive Deligne-Simpson problem in [ 9 ].”
• Normality of Marsden-Weinstein reductions for representations of quivers (2001, background): “Since X issim ple,the hom othety A !”
• Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity (2003, methodology): “Note that the method of [9] together with [7, Theorem 4.”
• Rigid modules over preprojective algebras (2005, unknown): “f modules over a preprojective algebra correspond to the elements of the canonical basis of a quantum group. These and many other results related to preprojective algebras can be found in [14], [15], [16], [17], [33], [37], [38], [39], [40], for example. 1.2. The aim of this article is twofold: Firs”