The No Gap Conjecture for tame hereditary algebras

Abstract

The "No Gap Conjecture" of Brüstle-Dupont-Pérotin 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.2010 Mathematics Subject Classification. 16G20; 20F55.

Lengths of maximal green sequences for tame path algebras

“…tilted algebras of type A [9]. Hermez-Igusa extended this result to cluster-tilted algebras of finite type and path algebras of tame type [11]. Therefore, for each Dynkin or extended Dynkin quiver Q, there are integers (Q) and (Q) such that…”

Quantum dilogarithm identities for n-cycle quivers

“…Roots from ΦpA ℓ q (and ΦpA j q) appear in Reineke order. On the other hand, if one of β or β 1 is in ΦpA ℓ q and the other is in ΦpA j q, then (8) β ă β 1 ùñ λpβ, β 1 q ď 0.…”

“…so that the longest roots r1, ℓs and r1 1 , j 1 s appear in the same column. One checks that again taking columns left to right yields an ordering satisfying (7) and (8).…”

“…The No Gap Conjecture [2, Conjecture 2.2] states that the set of lengths of all possible maximal green sequences forms an interval of integers. This has been proven up to tame (acyclic) types [8], but is still open for acyclic quivers, Γ n for example. There are maximal green sequences of length 8 for Γ 4 , e.g.…”

“…Recent progress has been made in classifying maximal green sequences. For example, the "No Gap Conjecture" of [2] has been established for tame hereditary algebras, and hence for quivers which are acyclic orientations of simply-laced extended Dynkin diagrams [8]. Furthermore, results on the lower bounds for lengths of maximal green sequences in certain types have appeared, e.g.…”