Known results

The convention, and a trap in the literature

Write b_ij for the number of arrows i → j minus the number j → i. To frame a quiver, add a frozen vertex i′ and one arrow i → i′ for each vertex i. A vertex is green when no arrow comes into it from a frozen vertex, that is, when its c-vector has no negative entry, and red when its c-vector has no positive entry. This is Brüstle, Dupont and Pérotin's convention, after Keller, and Igusa and Zhou use the same one, though they write their matrices differently.

Brüstle, Hermes, Igusa and Todorov use the opposite convention. Their framing amounts to ours on the quiver with every arrow reversed. So where their paper speaks of the source of a multiple arrow, this game speaks of its target, and their “target before source” is our “source before target”. Every ban on this site is stated in our convention. Counts are not affected: BDP Proposition 2.15 matches the MGS of a quiver with those of its opposite, length for length.

Finitely many: what is proved

• Finite (Dynkin) type. Finitely many MGS, and at least n of them, for any quiver mutation-equivalent to a Dynkin quiver (BDP Theorem 4.1). The game applies it to acyclic Dynkin quivers, since it does not decide mutation-equivalence.
• Acyclic Euclidean (tame) quivers. Finitely many, and at least one (BDP Theorem 5.2); more generally, finitely many for any quiver mutation-equivalent to an acyclic tame one (BHIT Theorem 4.0.5). The proof does not produce a length bound, so it cannot by itself turn a search into a complete list.
• Acyclic quivers with three vertices, wild ones included: finitely many, and at least one (BDP Theorem 5.4). Igusa and Zhou give a second proof (their Corollary 3.17).
• Two vertices. A₂ has exactly two MGS, of lengths 2 and 3; a connected two-vertex quiver with two or more arrows has exactly one, of length 2 (BDP Lemma 5.1).
• Shortest. For an acyclic quiver with n vertices the shortest MGS have length n: mutating sources one at a time always works (BDP Lemma 2.20 with Corollary 2.18).
• Longest, for Dynkin quivers. The number of positive roots: 6 for A₃, 12 for D₄, 36 for E₆ (BDP Theorem 4.4).
• Wild quivers with four or more vertices. No general finiteness theorem; Brüstle, Dupont and Pérotin left the question open. For 1 → 2 → 3 ⇒ 4 they counted 21 MGS, conditional on a conjecture of theirs; the engine certifies the same 21 by a complete walk that skips only what BHIT Corollary 3.3.2 rules out.

Totals for Dynkin quivers are on record too. The engine reproduces Brüstle, Dupont and Pérotin's published counts, for example 9 for 1 → 2 → 3 and 10 for 1 → 2 ← 3, 98 for linear A₄, and 253,085,705,387 for their E₆. Different orientations of the same diagram can have very different totals.

A proved ban: BHIT Corollary 3.3.2

The rule, in this game's convention: in any maximal green sequence of any quiver, no step mutates a vertex that is, at that moment, the target of two or more parallel arrows. It is Corollary 3.3.2 of Brüstle, Hermes, Igusa and Todorov, in a refereed journal paper, and it needs no hypotheses beyond what every quiver here satisfies.

A consequence for acyclic quivers (their Corollary 3.3.4): if there are two or more arrows i → j, every MGS mutates i for the first time before it first mutates j. Their paper calls it “target before source”, in their convention; in ours it reads source before target. Acyclicity matters here: they give a quiver with an oriented cycle where the order rule fails.

Example. In the Kronecker quiver, 1 ⇉ 2, vertex 2 starts as the target of the double arrow, so every MGS begins with 1. In fact (1, 2) is the only one.

How the game uses it. In the explorer a green vertex gets a ! mark when mutating it now would break the rule. You may still make the move; the panel of proved restrictions then says the prefix is proved forbidden. The engine's complete walks may skip such moves, which is how the Kronecker quiver's single MGS is certified: without the theorem, the search never finishes.

Featured: Igusa and Zhou on multiple edges (2019)

K. Igusa and Y. Zhou's paper Maximal green sequences of quivers with multiple edges (arXiv, 2019) asks what multiple arrows do to maximal green sequences. It is an unrefereed preprint: only version 1 exists, and we found no journal version when we checked on 5 November 2026. We feature it because it speaks directly to the question this game is built around, and we say plainly that the game shows its results but never relies on them.

The idea

Take an acyclic quiver Q with at least one multiple edge. Delete every multiple edge, and call the result Q′. Their Theorem 3.12 says that every MGS of Q is also an MGS of Q′, and its proof shows that it passes through the same c-matrices. Theorem 3.13 then says which MGS of Q′ survive in Q. In our reading: those that only ever use c-vectors whose support contains no multiple edge of Q, and that, for each multiple edge from i to j, never mutate a c-vector with j in its support while some green c-vector still has i in its support.

Consequences they draw (Corollaries 3.14 and 3.17):

• Q has at most as many MGS as Q′, and if Q′ has finitely many, so does Q;
• every acyclic quiver with at most three vertices has finitely many MGS, a new proof of BDP Theorem 5.4.

Quiver cutting

For quivers that may have oriented cycles they also prove a cutting result (Theorem 4.3): split the vertices into two parts, and suppose that every count of arrows between the parts is a multiple of some k greater than 1. Then every MGS of Q is an MGS of the two parts taken side by side, with the arrows between them deleted.

What we checked, and what we didn't

We tested their predictions, not their proofs. On six quivers, the MGS that Theorem 3.13 predicts are exactly the ones the engine certifies by complete search, including the worked example below. The proofs are brief in places, and we have not verified them.

Worked example: 1 ⇉ 2 → 3

Take the quiver 1 ⇉ 2 → 3: two arrows from 1 to 2, and one from 2 to 3. It is not in Igusa and Zhou's paper, which treats other examples, but their theorem applies to it, and it shows every idea on this page at small scale. Open it in the explorer and follow along.

1. What kind of quiver it is

Its Tits matrix is [[2, -2, 0], [-2, 2, -1], [0, -1, 2]]. The top-left 2 × 2 minor is 0 and the determinant is -2, so the form is indefinite and the quiver is wild, with only three vertices. Even so, it has finitely many MGS: it is acyclic with three vertices (BDP Theorem 5.4).

2. The proved ban

At the start, vertex 2 is the target of the double arrow, so by BHIT Corollary 3.3.2 no MGS begins with 2. Mutating 3 first reverses 2 → 3 but leaves 1 ⇉ 2 alone, so 2 is still the target and no MGS begins with (3, 2) either. Put another way (Corollary 3.3.4): every MGS mutates 1 before 2. In BHIT's own convention the same picture bans 1 before 2; the two statements say the same thing about mirror-image quivers.

3. Igusa and Zhou's prediction

Deleting the double edge leaves Q′: vertex 1 on its own, beside 2 → 3. Q′ has 7 MGS. No c-vector met along them has support containing both 1 and 2. In our reading of IZ Theorem 3.13, an MGS of Q′ survives in Q unless it mutates a c-vector containing e₂ while some green c-vector still contains e₁:

The prediction is exactly 3 MGS: (1, 2, 3), (1, 3, 2, 3), (3, 1, 2, 3).

4. The computation

The engine's complete search, skipping only the moves BHIT Corollary 3.3.2 rules out, finishes and finds exactly (1, 2, 3), (1, 3, 2, 3), (3, 1, 2, 3). That agrees with the preprint's prediction, and it is what makes this quiver's total of 3 certified.

5. Why 2 first is a dead end, though it is a legal move

Mutating 2 first is a perfectly good green move. Afterwards the c-vector of 1 is e₁+2e₂, whose support spans the double edge; IZ Theorem 3.12 would exclude it from every MGS. Keep alternating 2 and 1 and the mutated c-vectors run e₂, e₁+2e₂, 2e₁+3e₂, 3e₁+4e₂, 4e₁+5e₂, 5e₁+6e₂, 6e₁+7e₂, … and the vertices keep turning green again. A search without the theorem, to depth 25, finds no MGS starting with 2, but 2,060 branches were still going when it stopped: that is only “none found within the limits”. It is the theorem that turns it into “proved forbidden”. Whether every green continuation of (2) goes on forever has not been proved here, and the ban doesn't need it.

So among the green prefixes of length at most 2, the proved forbidden ones are those that start with 2, and (3, 2); every other one extends to an MGS. By the engine's complete search:

• (1): extends to an MGS
• (2): proved forbidden
• (3): extends to an MGS
• (1, 2): extends to an MGS
• (1, 3): extends to an MGS
• (2, 1): proved forbidden
• (2, 3): proved forbidden
• (3, 1): extends to an MGS
• (3, 2): proved forbidden

Now look for restrictions nobody has written down yet.

Still open, or not used

• No length bound for tame quivers. Finiteness is proved, but no explicit bound on the length of an MGS is known to the game, so a tame total is certified only if a complete search happens to finish. For Ã₂ drawn as 1 → 2 → 3 with 1 → 3, Brüstle, Dupont and Pérotin state that there are exactly five MGS; the engine finds the same five, but cannot certify the list, because some green branches are still going when its limits stop it.
• Wild quivers with four or more vertices: whether they have finitely many MGS is open in general.
• Lengths with gaps? Brüstle, Dupont and Pérotin conjectured that the lengths of the MGS of a quiver form an unbroken range. The game never uses that conjecture.
• Infinite green sequences are never claimed. A search that stops at its depth limit reports open branches, not endless ones.
• Mutation-equivalence is not computed. Theorems about quivers mutation-equivalent to Dynkin or tame ones are applied only to acyclic quivers, where that holds trivially.

References

1. T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences. Int. Math. Res. Not. IMRN 2014(16), 4547–4586. Cited here as BDP; their Appendix B tables are in arXiv version 1. doi:10.1093/imrn/rnt075 · arXiv:1205.2050
2. T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences. J. Algebra 473 (2017), 80–109. Cited here as BHIT. Uses the opposite framing convention to this game. doi:10.1016/j.jalgebra.2016.10.025 · arXiv:1503.07945
3. K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges. Preprint, 2019, version 1 only; not refereed. Cited here as IZ. arXiv:1902.07375

Sign-coherence of c-vectors, which green and red rely on, is due to Derksen, Weyman and Zelevinsky (J. Amer. Math. Soc. 23 (2010) 749–790), as cited by BDP. The engine also checks it at every step.