Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A deterministic algorithm for Harder-Narasimhan filtrations for representations of acyclic quivers

chi-yu Cheng

Abstract

Let $M$ be a representation of an acyclic quiver $Q$ over an infinite field $k$. We establish a deterministic algorithm for computing the Harder-Narasimhan filtration of $M$. The algorithm is polynomial in the dimensions of $M$, the weights that induce the Harder-Narasimhan filtration of $M$, and the number of paths in $Q$. As a direct application, we also show that when $k$ is algebraically closed and when $M$ is unstable, the same algorithm produces Kempf's maximally destabilizing one parameter subgroups for $M$.

A deterministic algorithm for Harder-Narasimhan filtrations for representations of acyclic quivers

Abstract

Let be a representation of an acyclic quiver over an infinite field . We establish a deterministic algorithm for computing the Harder-Narasimhan filtration of . The algorithm is polynomial in the dimensions of , the weights that induce the Harder-Narasimhan filtration of , and the number of paths in . As a direct application, we also show that when is algebraically closed and when is unstable, the same algorithm produces Kempf's maximally destabilizing one parameter subgroups for .

Paper Structure

This paper contains 22 sections, 22 theorems, 87 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. The main results
  3. Outline of the paper
  4. Acknowledgements
  5. Stability conditions for representations of quivers
  6. Set up
  7. Slope stability
  8. Weight stability
  9. The main results
  10. Completing Theorem B
  11. Instability in affine geometric invariant theory
  12. The quiver setting
  13. From the Harder-Narasimhan filtration to maximally destabilizing one parameter subgroups
  14. Maximizing Kempf function
  15. A short example
  16. ...and 7 more sections

Key Result

Theorem A

Let $\Theta$ be a weight and let $M$ be a representation of $Q$ with $\Theta(M) = 0$. For any weight $\kappa$ with $\kappa((\mathbf{Z}^{+})^{Q_0})>0$, if $\mu=\Theta/\kappa$ is the slope function, then any subrepresentation $M'$ with $\Theta(M')=\mathop{\mathrm{disc}}\nolimits(M,\Theta)$ contains th is the Harder-Narasimhan filtration of $M$ (with respect to $\mu$), then there is an $M_{l}$ in the

Theorems & Definitions (42)

  • Theorem A: \ref{['discrep_contains_1st_HN_term']}, \ref{['HN_contains_witness_discrepancy']}, \ref{['HN_contains_witness_discrepancy_corollary']}
  • Theorem B: \ref{['thmB_part_I']}, \ref{["one_PS_via_Kempf's_filtration"]}
  • Definition 2.1
  • Lemma 2.2: Lemma 2.1 MR1906875
  • Lemma 2.3: Lemma 2.2 MR1906875
  • Theorem 2.4: Theorem 2.5 MR1906875
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 32 more