Harmonic Norms on Quiver Representations over Non-Archimedean Fields
Abstract
Let K be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over K admits a split harmonic norm in the sense of Haiden--Katzarkov--Kontsevich--Pandit (HKKP), answering a question from their preprint \emph{Towards Categorical Kähler Geometry}. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope--rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.
Related
- Diagrams for Roots and Quiver Representations in the context of the Gabriel-Kac Theorem
- An Extended Generalization of RSK Correspondence via A Type Quiver Representations
- Flag Varieties of General Quiver Representations: Characteristic Classes and Semi-Invariants
- A Mixed Robinson--Schensted--Knuth Construction in the Representation Theory of GLn over Local non-Archimedean Fields
- Quiver Bases of Cartan Squares of Minuscule Representations
BibTeX
@misc{benbassat2026harmonic,
title = {{Harmonic Norms on Quiver Representations over Non-Archimedean Fields}},
author = {Oren Ben-Bassat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01694},
eprint = {2610.01694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01694},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Harmonic Norms on Quiver Representations over Non-Archimedean Fields
AU - Oren Ben-Bassat
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01694
UR - https://arxiv.org/abs/2610.01694
AB - Let K be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over K admits a split harmonic norm in the sense of Haiden--Katzarkov--Kontsevich--Pandit (HKKP), answering a question from their preprint \emph{Towards Categorical Kähler Geometry}. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope--rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.
ER -