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Algorithmic Aspects of Semistability of Quiver Representations

Authors Yuni Iwamasa , Taihei Oki , Tasuku Soma

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ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Mathematics of computing → Submodular optimization and polymatroids
Keywords
  • quivers
  • σ-semistability
  • King’s criterion
  • operator scaling
  • submodular flow

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Abstract

We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and σ-semistability, finding the maximizers of King’s criterion, and computing the Harder-Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King’s criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the σ-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.

Cite As Get BibTex

Yuni Iwamasa, Taihei Oki, and Tasuku Soma. Algorithmic Aspects of Semistability of Quiver Representations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 99:1-99:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.99

Author Details

Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Japan
Taihei Oki
  • Institute for Chemical Reaction Design and Discovery (ICReDD), Hokkaido University, Sapporo, Japan
  • Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan
Tasuku Soma
  • The Institute of Statistical Mathematics, Tokyo, Japan
  • Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan

Funding

This work was supported by JSPS KAKENHI Grant Number JP24K21315, Japan.
  • Iwamasa, Yuni: Supported by JSPS KAKENHI Grant Numbers JP22K17854, JP24K02901, Japan.
  • Oki, Taihei: Supported by JST ERATO Grant Number JPMJER1903, JST CREST Grant Number JPMJCR24Q2, JST FOREST Grant Number JPMJFR232L, JSPS KAKENHI Grant Number JP22K17853, and Start-up Research Funds in ICReDD, Hokkaido University.
  • Soma, Tasuku: Supported by JPSP KAKENHI Grant Number JP19K20212 and JST, PRESTO Grant Number JPMJPR24K5, Japan.

Acknowledgements

The authors thank Hiroshi Hirai and Keiya Sakabe for their valuable comments on an earlier version of this paper. The last author thanks Cole Franks for bringing the submodularity of quiver representations to his attention. The authors thank an anonymous reviewer for pointing out the reference [Mulmuley, 2017].

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