The Deligne–Simpson problem—a survey
By Vladimir Petrov Kostov. 2002
66 citations recorded by the source corpus.
Abstract The Deligne–Simpson problem (DSP) (respectively the weak DSP) is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes C j ⊂ GL ( n , C ) or c j ⊂ gl ( n , C ) so that there exist irreducible (respectively with trivial centralizer) ( p + 1 ) -tuples of matrices M j ∈ C j or A j ∈ c j satisfying the equality M 1 ⋯ M p + 1 = I or A 1 + ⋯ + A p + 1 = 0 . The matrices M j and A j are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. The present paper offers a survey of the results known up to now concerning the DSP.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d41a32aa2c76401a3adfaf2c06faf3afe64c35cb
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In Algebra and Number Theory
- P. Shaw
- W. Crawley‐Boevey
- Daisuke Yamakawa
- Carlos Simpson
- P. Boalch
- Vladimir Petrov Kostov
- V. Ginzburg
- I. Losev
In Discrete Mathematics and Combinatorics
- P. Shaw
- W. Crawley‐Boevey
- Daisuke Yamakawa
- Vladimir Petrov Kostov
- Yuya Mizuno
- Markus Reineke
- Bernard Leclerc
- Tamás Hausel
In Mathematical Physics
- W. Crawley‐Boevey
- P. Shaw
- P. Boalch
- Daisuke Yamakawa
- Carlos Simpson
- H. Nakajima
- I. Losev
- Markus Reineke
Citations in the corpus
- Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem (2004, background): “[11] V. P. Kostov, On the Deligne-Simpson problem, C. R. Acad.”
- Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity (2003, unknown): “Now A has type ξ if and only if for all λ ∈ K, the Jordan normal form of A only involves Jordan blocks with eigenvalue λ up to size rλ, where rλ = |{ℓ | 1 ≤ ℓ ≤ d, ξi = λ}|.”
- Arithmetic harmonic analysis on character and quiver varieties II (2008, unknown): “Note that when g = 0 the quantity −2∆(μ) is Katz’s index of rigidity of a solution to X1 · · · Xk = I with Xi ∈ Ci (see for example [19][p.”