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Harmonic norms on quiver representations without spherical completeness (Haiden-Katzarkov-Kontsevich-Pandit Question 6.24)

Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagonalise.

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algebraOct 1, 2026Significance 8/100Registry: unreviewed

Harmonic norms on quiver representations without spherical completeness (Haiden-Katzarkov-Kontsevich-Pandit Question 6.24)

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Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagona…

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Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagonalise.

Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagonalise.

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Harmonic norms on quiver representations without spherical completeness (Haiden-Katzarkov-Kontsevich-Pandit Question 6.24) — Mathematical Frontier Network