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This is a Quantitative Study study.

October 13, 2025Open Access

The tame Deligne-Simpson problem

Key Points

Proving necessity in Crawley-Boevey's conjecture for the deligne-simpson problem has substantial implications for representation theory.
Key findings emphasize that simple representations connect directly to lower-dimensional representations within conjugacy classes.
The methodology leverages nonabelian hodge correspondence and variations of parabolic weights to simplify complex scenarios.
The results suggest that understanding affine dynkin diagrams could lead to broader applications in representation theory.

Abstract

The objective of this article is to prove the necessity statement in Crawley-Boevey's conjectural solution to the (tame) Deligne-Simpson problem. We use the nonabelian Hodge correspondence, variation of parabolic weights and results of Schedler-Tirelli to reduce to simpler situations, where every conjugacy class is semi-simple and the underlying quiver is (1) an affine Dynkin diagram or (2) an affine Dynkin diagram with an extra vertex. In case (1), a nonexistence result of Kostov applies. In case (2), the key step is to show that simple representations, if exist, lie in the same connected component as direct sums of lower dimensional ones.

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Cite This Study

Shu Cheng (Mon,) studied this question.

synapsesocial.com/papers/68ecfebf950606aabec0960d https://doi.org/https://doi.org/10.48550/arxiv.2509.11841

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