Quiver Quest

Choose a quiver

Mutate green vertices until every vertex is red: that is a maximal green sequence. Everything you try stays in your browser until you choose to submit. New here? Take the tutorial, or start with A₂.

Two spotted tabby kittens with blue eyes snuggled together against a grey stone wall. — Photo by Abhishekgowda28, CC0 1.0

Dynkin mode

Curious Kitten

Finite type. Every quiver here has a certified total, so you can see exactly how many sequences there are to find.

A₂
type A₂ · 2 vertices
Arrows: 1 → 2
Two vertices, one arrow. The smallest place to learn what a mutation does.
Enumeration complete and certified
Exactly 2 maximal green sequences. Brüstle–Dupont–Pérotin's Lemma 5.1 states that A₂ has exactly two, of lengths 2 and 3. The engine's complete walk of the exchange graph agrees.
Lengths
Number of maximal green sequences of each length
Length MGS
2 1
3 1
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- It has exactly two MGS, of lengths 2 and 3. (Brüstle–Dupont–Pérotin 2014, Lemma 5.1)
- The shortest MGS have length 2, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 3, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 5.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore A₂
A₃, in a line
type A₃ · 3 vertices
Arrows: 1 → 2, 2 → 3
Three vertices in a row, every arrow pointing the same way.
Enumeration complete and certified
Exactly 9 maximal green sequences. Brüstle–Dupont–Pérotin's tables (arXiv:1205.2050v1, Appendix B, Figure 22) give 9 for this quiver, with these lengths. The engine's complete walk of the exchange graph agrees.
Lengths
Number of maximal green sequences of each length
Length MGS
3 1
4 4
5 2
6 2
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 6, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore A₃, in a line
A₃, a sink in the middle
type A₃ · 3 vertices
Arrows: 1 → 2, 3 → 2
The same graph as before, but both arrows point inward. Does the orientation matter?
Enumeration complete and certified
Exactly 10 maximal green sequences. Brüstle–Dupont–Pérotin's tables (arXiv:1205.2050v1, Appendix B, Figure 22) give 10 for this quiver, with these lengths. The engine's complete walk of the exchange graph agrees.
Lengths
Number of maximal green sequences of each length
Length MGS
3 2
4 2
5 2
6 4
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 6, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore A₃, a sink in the middle
A₄, in a line
type A₄ · 4 vertices
Arrows: 1 → 2, 2 → 3, 3 → 4
One vertex longer.
Enumeration complete and certified
Exactly 98 maximal green sequences. Brüstle–Dupont–Pérotin's tables (arXiv:1205.2050v1, Appendix B, Figure 23) give 98 for this quiver, with these lengths. The engine's complete walk of the exchange graph agrees.
Lengths
Number of maximal green sequences of each length
Length MGS
4 1
5 10
6 22
7 22
8 18
9 13
10 12
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 4, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 10, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore A₄, in a line
D₄
type D₄ · 4 vertices
Arrows: 1 → 2, 3 → 2, 4 → 2
Three arms meeting at a centre.
Enumeration complete and certified
Exactly 468 maximal green sequences. Brüstle–Dupont–Pérotin's tables (arXiv:1205.2050v1, Appendix B, Figure 25) give 468, with these lengths, for the opposite orientation, arrows out of the centre; their Proposition 2.15 matches the MGS of a quiver with those of its opposite, length for length. The engine's complete walk of the exchange graph on this orientation agrees.
Lengths
Number of maximal green sequences of each length
Length MGS
4 6
5 6
6 6
7 18
8 24
9 24
10 24
11 144
12 216
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 4, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 12, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore D₄
A₅, zigzag
type A₅ · 5 vertices
Arrows: 1 → 2, 3 → 2, 3 → 4, 5 → 4
Arrows alternating direction along a line of five.
Enumeration complete and certified
Exactly 19,438 maximal green sequences. Not in Brüstle–Dupont–Pérotin's tables, which list linear A₅. The engine's complete walk of the exchange graph gives this count, and listing all 19,438 sequences one by one gives the same.
Lengths
Number of maximal green sequences of each length
Length MGS
5 16
6 38
7 72
8 134
9 200
10 376
11 780
12 1,644
13 3,406
14 4,876
15 7,896
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 5, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 15, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore A₅, zigzag
D₅
type D₅ · 5 vertices
Arrows: 1 → 2, 2 → 3, 4 → 3, 5 → 3
A longer arm on the D shape.
Enumeration complete and certified
Exactly 435,864 maximal green sequences. Not in Brüstle–Dupont–Pérotin's tables. From the engine's complete walk of the exchange graph alone, the method that reproduces Brüstle–Dupont–Pérotin's published counts for the A₃, A₄ and D₄ classes, linear A₅ and A₆, E₆ and E₇.
Lengths
Number of maximal green sequences of each length
Length MGS
5 12
6 46
7 52
8 92
9 206
10 310
11 546
12 1,178
13 2,326
14 3,696
15 6,684
16 10,772
17 30,700
18 87,212
19 153,216
20 138,816
Known facts
- Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
- The shortest MGS have length 5, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The longest MGS have length 20, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4
Explore D₅

Number of maximal green sequences of each length
Length	MGS
2	1
3	1

Number of maximal green sequences of each length
Length	MGS
3	1
4	4
5	2
6	2

Number of maximal green sequences of each length
Length	MGS
3	2
4	2
5	2
6	4

Number of maximal green sequences of each length
Length	MGS
4	1
5	10
6	22
7	22
8	18
9	13
10	12

Number of maximal green sequences of each length
Length	MGS
4	6
5	6
6	6
7	18
8	24
9	24
10	24
11	144
12	216

Number of maximal green sequences of each length
Length	MGS
5	16
6	38
7	72
8	134
9	200
10	376
11	780
12	1,644
13	3,406
14	4,876
15	7,896

Number of maximal green sequences of each length
Length	MGS
5	12
6	46
7	52
8	92
9	206
10	310
11	546
12	1,178
13	2,326
14	3,696
15	6,684
16	10,772
17	30,700
18	87,212
19	153,216
20	138,816

E₆

type E₆ · 6 vertices

Arrows: 1 → 2, 2 → 3, 4 → 3, 5 → 4, 6 → 3

The first exceptional type. The largest Dynkin quiver here.

Enumeration complete and certified

Exactly 2,929,361,906,674 maximal green sequences. Not in Brüstle–Dupont–Pérotin's tables: their E₆ count, 253,085,705,387, is for a different orientation. From the engine's complete walk of the exchange graph alone, the method that reproduces Brüstle–Dupont–Pérotin's published counts for the A₃, A₄ and D₄ classes, linear A₅ and A₆, E₆ and E₇.

Lengths

Number of maximal green sequences of each length
Length	MGS
6	30
7	216
8	606
9	810
10	1,662
11	3,550
12	7,288
13	15,996
14	32,436
15	69,716
16	143,020
17	293,598
18	673,070
19	1,759,942
20	4,414,466
21	9,827,664
22	19,761,982
23	40,966,380
24	97,521,134
25	244,062,414
26	621,621,704
27	1,667,371,918
28	4,688,376,182
29	13,049,807,538
30	36,829,941,184
31	102,212,006,280
32	257,239,709,472
33	518,397,612,352
34	777,288,110,144
35	778,731,443,520
36	438,216,350,400

Known facts

Proved: finitely many MGS, as for every quiver of finite (Dynkin) type. (Brüstle–Dupont–Pérotin 2014, Theorem 4.1)
The shortest MGS have length 6, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
The longest MGS have length 36, the number of positive roots. (Brüstle–Dupont–Pérotin 2014, Theorem 4.4)

Full references

T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.1
T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 4.4

Explore E₆

A calico cat sitting on top of a wooden fence in front of a leafy green bush, eyes half closed. — Photo by TechnoKittyCat, CC0 1.0

Tame mode

House Cat

Euclidean type. Finitely many maximal green sequences is a theorem, but for most of these nobody has a complete list.

The Kronecker quiver
type Ã₁ · 2 vertices
Arrows: 1 → 2 (×2)
Two vertices and a double arrow. Small, but its representations come in infinite families.
Enumeration complete and certified
Exactly 1 maximal green sequence. Brüstle–Dupont–Pérotin's Lemma 5.1 states exactly one. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, agrees; without the theorem the walk does not finish.
Lengths
Number of maximal green sequences of each length
Length MGS
2 1
Known facts
- Proved: finitely many MGS, as for every acyclic Euclidean quiver. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
- It has exactly one MGS, of length 2. (Brüstle–Dupont–Pérotin 2014, Lemma 5.1)
- The shortest MGS have length 2, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.2
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 5.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore The Kronecker quiver
Ã₂, an acyclic triangle
type Ã₂ · 3 vertices
Arrows: 1 → 2, 2 → 3, 1 → 3
A triangle whose arrows never go all the way round.
Finiteness proved
The total is finite, but not known: nobody has a certified complete list.
Known facts
- Proved: finitely many MGS, as for every acyclic Euclidean quiver. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The proof of finiteness gives no upper bound on the length of an MGS, and the game knows of none. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.2
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ã₂, an acyclic triangle
Ã₃, a square
type Ã₃ · 4 vertices
Arrows: 1 → 2, 2 → 3, 1 → 4, 4 → 3
Two paths from vertex 1 to vertex 3.
Finiteness proved
The total is finite, but not known: nobody has a certified complete list.
Known facts
- Proved: finitely many MGS, as for every acyclic Euclidean quiver. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
- The shortest MGS have length 4, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The proof of finiteness gives no upper bound on the length of an MGS, and the game knows of none. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.2
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ã₃, a square
D̃₄, four arms
type D̃₄ · 5 vertices
Arrows: 1 → 5, 2 → 5, 3 → 5, 4 → 5
A centre with four arms, every arrow pointing in.
Finiteness proved
The total is finite, but not known: nobody has a certified complete list.
Known facts
- Proved: finitely many MGS, as for every acyclic Euclidean quiver. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
- The shortest MGS have length 5, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- The proof of finiteness gives no upper bound on the length of an MGS, and the game knows of none. (Brüstle–Dupont–Pérotin 2014, Theorem 5.2)
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.2
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore D̃₄, four arms

Number of maximal green sequences of each length
Length	MGS
2	1

A Canada lynx with a thick grey winter coat sitting in front of dark conifers, with snow on the ground. — Photo by Dominique Bérubé (berdom on iNaturalist), CC0 1.0

Wild mode

Wild Cat Researcher

Wild type, the research laboratory. Most here are almost tame, one vertex beyond a tame quiver. A few small ones have certified totals; for others even finiteness is open. Look for sequences, and for beginnings that can never be finished.

Almost tame: one vertex beyond tame

Each of these is a tame quiver plus one vertex. Delete that vertex and you are back in tame country; delete any vertex and what is left is Dynkin or tame. They are the wild quivers closest to the ones we understand, so a restriction found in tame mode is most likely to survive, or to break interestingly, here.

1 ⇉ 2 → 3
wild type · 3 vertices
Arrows: 1 → 2 (×2), 2 → 3
A double arrow followed by a single one: the game's worked example of a proved ban. See Known Results.
Enumeration complete and certified
Exactly 3 maximal green sequences. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, finds exactly three. Igusa–Zhou's Theorem 3.13 (an unrefereed preprint) predicts the same three; see Known Results.
Lengths
Number of maximal green sequences of each length
Length MGS
3 1
4 2
Known facts
- Proved: finitely many MGS, as for every acyclic quiver with three vertices. (Brüstle–Dupont–Pérotin 2014, Theorem 5.4)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.4
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore 1 ⇉ 2 → 3
1 ⇉ 2 ← 3
wild type · 3 vertices
Arrows: 1 → 2 (×2), 3 → 2
The Kronecker quiver with one more vertex pointing into its sink.
Enumeration complete and certified
Exactly 3 maximal green sequences. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, finds exactly three.
Lengths
Number of maximal green sequences of each length
Length MGS
3 2
4 1
Known facts
- Proved: finitely many MGS, as for every acyclic quiver with three vertices. (Brüstle–Dupont–Pérotin 2014, Theorem 5.4)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.4
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore 1 ⇉ 2 ← 3
3 → 1 ⇉ 2
wild type · 3 vertices
Arrows: 3 → 1, 1 → 2 (×2)
The Kronecker quiver with one more vertex feeding its source.
Enumeration complete and certified
Exactly 3 maximal green sequences. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, finds exactly three.
Lengths
Number of maximal green sequences of each length
Length MGS
3 1
4 2
Known facts
- Proved: finitely many MGS, as for every acyclic quiver with three vertices. (Brüstle–Dupont–Pérotin 2014, Theorem 5.4)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.4
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore 3 → 1 ⇉ 2
3 ← 1 ⇉ 2
wild type · 3 vertices
Arrows: 1 → 2 (×2), 1 → 3
The Kronecker quiver whose source also sends an arrow to a new vertex.
Enumeration complete and certified
Exactly 3 maximal green sequences. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, finds exactly three.
Lengths
Number of maximal green sequences of each length
Length MGS
3 2
4 1
Known facts
- Proved: finitely many MGS, as for every acyclic quiver with three vertices. (Brüstle–Dupont–Pérotin 2014, Theorem 5.4)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.4
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore 3 ← 1 ⇉ 2
1 ⇉ 2 ⇉ 3
wild type · 3 vertices
Arrows: 1 → 2 (×2), 2 → 3 (×2)
Two double arrows in a row.
Enumeration complete and certified
Exactly 1 maximal green sequence. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, finds exactly one.
Lengths
Number of maximal green sequences of each length
Length MGS
3 1
Known facts
- Proved: finitely many MGS, as for every acyclic quiver with three vertices. (Brüstle–Dupont–Pérotin 2014, Theorem 5.4)
- The shortest MGS have length 3, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 2 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Every MGS mutates 2 for the first time before it first mutates 3, because of the 2 arrows 2 → 3. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Theorem 5.4
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore 1 ⇉ 2 ⇉ 3
Ã₂ with a tail
wild type · 4 vertices
Arrows: 1 → 2, 2 → 3, 1 → 3, 3 → 4
The acyclic triangle with one vertex hanging off it.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 4, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ã₂ with a tail
Ã₃ with a tail
wild type · 5 vertices
Arrows: 1 → 2, 2 → 3, 1 → 4, 4 → 3, 3 → 5
The square, plus a vertex leaving its sink.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 5, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ã₃ with a tail
D̃₄ with a longer arm
wild type · 6 vertices
Arrows: 1 → 5, 2 → 5, 3 → 5, 4 → 5, 6 → 1
Four arms into a centre, one of them a vertex longer.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 6, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore D̃₄ with a longer arm
Five-armed star
wild type · 6 vertices
Arrows: 1 → 6, 2 → 6, 3 → 6, 4 → 6, 5 → 6
One arm more than D̃₄ tips the balance into wild.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 6, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Five-armed star
Ẽ₆ plus one (T₃,₃,₄)
wild type · 8 vertices
Arrows: 3 → 2, 2 → 1, 5 → 4, 4 → 1, 8 → 7, 7 → 6, 6 → 1
Ẽ₆'s three equal arms, with one arm a vertex longer.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 8, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ẽ₆ plus one (T₃,₃,₄)
Ẽ₇ plus one (T₂,₄,₅)
wild type · 9 vertices
Arrows: 2 → 1, 5 → 4, 4 → 3, 3 → 1, 9 → 8, 8 → 7, 7 → 6, 6 → 1
Ẽ₇ with one arm stretched by a vertex.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 9, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore Ẽ₇ plus one (T₂,₄,₅)
E₁₀: Ẽ₈ plus one (T₂,₃,₇)
wild type · 10 vertices
Arrows: 2 → 1, 4 → 3, 3 → 1, 10 → 9, 9 → 8, 8 → 7, 7 → 6, 6 → 5, 5 → 1
Ẽ₈'s long arm, one vertex longer: the hyperbolic E₁₀.
Checking the record…
Known facts
- Open: no theorem the game applies decides whether this quiver has finitely many MGS.
- The shortest MGS have length 10, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- No theorem the game applies bounds the length of an MGS from above.
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
Explore E₁₀: Ẽ₈ plus one (T₂,₃,₇)

Number of maximal green sequences of each length
Length	MGS
3	1
4	2

Number of maximal green sequences of each length
Length	MGS
3	2
4	1

Number of maximal green sequences of each length
Length	MGS
3	1
4	2

Number of maximal green sequences of each length
Length	MGS
3	2
4	1

Number of maximal green sequences of each length
Length	MGS
3	1

Further out

Wild by more than one vertex's worth.

Three-arrow Kronecker
wild type · 2 vertices
Arrows: 1 → 2 (×3)
Two vertices and a triple arrow: one arrow, not one vertex, beyond the Kronecker quiver.
Enumeration complete and certified
Exactly 1 maximal green sequence. Brüstle–Dupont–Pérotin's Lemma 5.1 states exactly one. The engine's complete walk of the exchange graph, skipping only the mutations Brüstle–Hermes–Igusa–Todorov's Corollary 3.3.2 rules out, agrees; without the theorem the walk does not finish.
Lengths
Number of maximal green sequences of each length
Length MGS
2 1
Known facts
- Proved: exactly one MGS, as for every two-vertex quiver with two or more arrows. (Brüstle–Dupont–Pérotin 2014, Lemma 5.1)
- It has exactly one MGS, of length 2. (Brüstle–Dupont–Pérotin 2014, Lemma 5.1)
- The shortest MGS have length 2, the number of vertices: mutating sources one at a time always works. (Brüstle–Dupont–Pérotin 2014, Lemma 2.20)
- Every MGS mutates 1 for the first time before it first mutates 2, because of the 3 arrows 1 → 2. (In this game's convention; see Known Results.) (Brüstle–Hermes–Igusa–Todorov 2017, Corollary 3.3.4)
- Igusa and Zhou's results on multiple edges apply too. Their paper is an unrefereed preprint, so the game shows them but never relies on them. (Igusa–Zhou 2019, Theorems 3.12–3.13 (unrefereed preprint))
Full references
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 5.1
- T. Brüstle, G. Dupont, M. Pérotin, On maximal green sequences, IMRN 2014(16) 4547–4586, doi:10.1093/imrn/rnt075, arXiv:1205.2050, Lemma 2.20
- T. Brüstle, S. Hermes, K. Igusa, G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra 473 (2017) 80–109, doi:10.1016/j.jalgebra.2016.10.025, arXiv:1503.07945, Corollary 3.3.4
- K. Igusa, Y. Zhou, Maximal green sequences of quivers with multiple edges, arXiv:1902.07375v1 (2019), preprint, no journal version found, Theorems 3.12–3.13, Corollary 3.14
Explore Three-arrow Kronecker