The No Gap Conjecture for tame hereditary algebras

TL;DR

This paper proves the No Gap Conjecture for tame hereditary algebras, showing that all maximal green sequences can be polygonally deformed into each other, implying a continuous spectrum of sequence lengths.

Contribution

It establishes the stronger conjecture that any two maximal green sequences are polygonally deformable for all tame hereditary algebras, confirming the No Gap Conjecture in this case.

Findings

No gaps in lengths of maximal green sequences for tame hereditary algebras

All maximal green sequences are polygonally deformable into each other

The result holds over any algebraically closed field

Abstract

The "No Gap Conjecture" of Br\"ustle-Dupont-P\'erotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be "polygonally deformed" into each other. We prove this stronger conjecture for all tame hereditary algebras over any field.