Artin Group Presentations Arising from Cluster Algebras
Jacob Haley1

arXiv:1410.3530v2 [] 18 Dec 2017

University of Notre Dame, Department of Mathematics
Notre Dame IN 46556

David Hemminger2
Duke University, Mathematics Department
Box 90320, Durham NC 27708-0320

Aaron Landesman3
Harvard University, Department of Mathematics
One Oxford Street, Cambridge MA 02138

Hailee Peck4
Millikin University, Department of Mathematics
1184 W Main St., Decatur IL 62522

Abstract
In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can
be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined
the presentation of a reflection group associated to a Dynkin diagram in
terms of an edge-weighted, oriented graph, and proved that this group is
invariant (up to isomorphism) under diagram mutations. In this paper, we
extend Barot and Marsh’s results to Artin group presentations, defining new
generator relations and showing mutation-invariance for these presentations.
Keywords: Artin group, cluster algebra, diagram mutations

1

jachaley@umich.edu
dhemminger@math.ucla.edu
3
aaronlandesman@college.harvard.com
4
hpeck@millikin.edu
2

1. Introduction & Motivation
In [FZ02], Fomin and Zelevinsky first introduced the concept of cluster algebras. Barot and Marsh extended Fomin and Zelevinsky’s results
in [BM13], providing a presentation of the reflection group associated to a
Dynkin diagram with generators that correspond to elements of a companion basis associated to a seed of a finite type cluster algebra. They define
generator relations corresponding to chordless cycles arising in diagrams of
finite type in order to give a Coxeter group presentation for these diagrams.
They also proved that this Coxeter group presentation is invariant up to
isomorphism under the mutation equivalence relation. That is, given a diagram Γ and a diagram mutation equivalent to Γ, denoted Γ0 = µk (Γ), they
proved that WΓ ∼
= WΓ0 , where WΓ and WΓ0 are the group presentations
corresponding to Γ and Γ0 , respectively.
In our paper, we define AΓ to be the Artin group presentation arising
from a cluster algebra, where Γ is the diagram associated to the cluster
algebra. We provide the necessary relations for the generators of the group,
and show that these relations hold under mutations of vertices in a diagram.
Our main result is to show that this Artin group presentation is invariant,
up to isomorphism, under the mutation equivalence relation. We state the
result here, but present the detailed proof in Section 6.
Theorem 1.1 (Theorem 6.1). Let Γ be a diagram of finite type, and let
Γ0 = µk (Γ) be the mutation of Γ at vertex k. Then AΓ ∼
= AΓ0 .
Section 2 provides the necessary definitions and fundamental results from
[BM13] to motivate our own results. For further definitions and references
on the topic, we refer the reader to [FZ02]. Section 3 will review theory
from [FZ02], [FZ03] as well as review the classifications (from [BM13]) of
mutations of diagrams and their oriented chordless cycles. In Section 4, we
define the appropriate relations for our Artin group presentations. Section
5 specifies how certain relations in chordless cycles imply other relations in
those chordless cycles. Section 6 will provide the proof that the Artin group
defined for a diagram Γ is invariant up to isomorphism under mutations of
Γ. Finally, in Section 7, we will state a conjecture which could extend the
main result to diagrams of affine type.
In the process of writing this paper, we became aware of related results
obtained independently by Grant and Marsh in [GM14] and Qiu in [Qiu14].
Using different approaches, the two papers found relations for the Artin
group in simply-laced diagrams. However, their relations differ from the
2

ones used in our paper, and we show that our results hold for all diagrams of
finite type. Furthermore, our approach to the problem is more combinatorial
than the topological approaches found in [Qiu14] and [GM14].
2. Background
We begin by introducing some preliminary notations and definitions
which will aid the reader in understanding the results in the following sections. For further references on cluster algebras, we refer the reader to
[FZ02, FZ03], and for a more detailed description of Artin group presentations, we direct attention to [FN61]. We also provide references to several
lemmas and propositions from [BM13] which were helpful in formulating our
own results.
The initial introduction of cluster algebras by Fomin and Zelevinsky was
aimed at making further strides in the areas of representation theory, Lie
theory, and total positivity. Since then, the study of cluster algebras has
provided a motivation for applications in various other areas of mathematics, including quiver representations. Of particular interest were finite type
cluster algebras, those with a finite number of distinguished generators. In
the sequel to their introductory paper ([FZ03]), Fomin and Zelevinsky introduce the concept of mutation equivalence between diagrams, proving that
a connected graph is mutation equivalent to an oriented Dynkin diagram if
and only if all mutation equivalent graphs have edge weights not exceeding
3. In particular, this proves that finite type cluster algebras can be classified
by Dynkin diagrams.
A cluster algebra is an integral domain which can be generated by a set
of elements called cluster variables that satisfy certain exchange relations.
Following the style of [FZ02] and [BM13], we will define cluster algebras in
terms of skew-symmetrisable matrices (that is, a matrix B such that there
exists a diagonal matrix D of the same size with Dii > 0 such that DB is
skew-symmetric). Let F = Q(u1 , u2 , . . . , un ) be the field of rational functions
in n indeterminates over Q. We will define an initial seed for the cluster
algebra to be a fixed pair (x, B), where x = {x1 , . . . , xn } is a free generating
set of F and B is an n × n skew-symmetric matrix. Define x0k ∈ F by the
exchange relation
Y B
Y −B
x0k xk =
xi ik +
xi ik
Bik >0

Bik <0

3

Then, given an initial seed (x, B) and k ∈ 1, 2, . . . , n, we can define a mutation of the seed at k, denoted µk (x, B) = (x0 , B 0 ) where:
(
−Bij
if i = k or j = k;
0
Bij =
|Bik |Bkj +Bik |Bkj |
Bij +
otherwise.
2
and x0 = {x1 , x2 , . . . , xk−1 , x0k , xk+1 , . . . , xn }. Such a mutation or a sequence
of such mutations generate seeds which in turn generate all cluster variables
in that, for each x = {x1 , . . . , xn } corresponding to a seed of the cluster
algebra, the entries xi are the cluster variables.
A cluster algebra is said to be of finite type if the number of cluster variables that generate it is finite (if it has finitely many seeds). For
each finite type cluster algebra, we can associate to its corresponding skewsymmetrisable matrix an edge-weighted, oriented graph, called a diagram.
We will often denote this diagram by Γ, and the vertex set of Γ by V (Γ). We
will denote two connected vertices by i → j, or by i − j if the orientation is
not specified. The diagram is determined by the following: for i, j ∈ V (Γ),
w
i−
→ j if and only if Bij > 0 and w = |Bij Bji | is the weight of the edge. A
skew-symmetrisable matrix B is 2-finite if |Bij Bji | ≤ 3 for i, j ∈ {1, . . . , n}.
By [FZ02, 7.5], we have that if B is 2-finite, all 3-cycles in the unoriented
graph underlying our diagram must be oriented cyclically.
Just as we can define mutations of the seeds of a cluster variable, we can
also define mutations of a diagram associated to a cluster algebra of finite
type by the following set of rules:
Proposition 2.1. [FZ03] Let B be a 2-finite skew-symmetrisable matrix.
Then Γ(µk (B)) is uniquely determined by Γ(B) as follows:
• Reverse the orientations of all edges in Γ(B) incident to k (leaving the
weights unchanged)
a

b

• For any path in Γ(B) of form i −
→ k →
− j (i.e. with a, b positive),
let c be the weight on the edge j → i, taken to be zero if√there is
√ no
√
0
0
0
such arrow. Let c be determined by c √
≥ 0 and ± c ± c = ab,
√
where the sign before c (respectively, c0 ) is positive if the arrows
form an oriented cycle and negative otherwise. Then Γ(B) changes as
in Figure 1, taking the case c0 = 0 to mean no arrow between i and j.

4

k
•
a
•
i

k
•
a

b
•

c

•
j

i

b
c’

•
j

Figure 1: Mutation of Γ at a node k

Notation 2.2. We notate this mutation of Γ(B) at vertex k by µk (Γ).
Notation 2.3. Given a diagram Γ, Barot and Marsh define for i, j ∈ V (Γ),


2 if i and j are not connected;



3 if i and j are connected by an edge of weight 1;
mij =

4 if i and j are connected by an edge of weight 2;



6 if i and j are connected by an edge of weight 3.
Then, they define W (Γ) to be the group generated by si , for i ∈ V (Γ),
under the following relations. Note that e will denote the identity element
of W (Γ).
1. s2i = e for all i;
2. (si sj )mij = e for all i 6= j;
3. For any chordless cycle (as defined in Definition 3.1) C in Γ, where
w

w

wd−1

w

1
2
0
C = i0 −→
i1 −→
· · · −−−→ id−1 −→
i0

and all of the weights wk are 1 or w0 = 2, we have
(si0 si1 · · · sid−2 sid−1 sid−2 · · · si1 )2 = e.
Using this group presentation, Barot and Marsh state the following result:
Theorem 2.4. [BM13, Theorem A] Let Γ be the diagram associated to a
seed in a cluster algebra of finite type. Then W (Γ) is isomorphic to the
corresponding reflection group.
5

In Section 3 of [BM13], Barot and Marsh provide an alteration of the
group W (Γ) in order to extend the group definition to any diagram of finite
type. More specifically, they provide relations such that WΓ , as defined
below, is isomorphic to W (Γ) ([BM13, Proposition 4.5]).
Definition 2.5. Let WΓ be the group with generators si , i = 1, 2, . . . , n,
subject to the following relations:
• (R1) s2i = e for all i
• (R2) (si sj )mij = e for all i 6= j
Furthermore, for a chordless cycle C : i0 → i1 → · · · → id−1 → i0 and for
a = 0, 1, 2, . . . , d − 1, define r (ia , ia+1 ) = sia sia+1 · · · sia+d−1 sia+d−2 · · · sia+1 .
Then we have the following relations:
• (R3)(a) If all the weights in the edges of C are 1, then r(ia , ia+1 )2 = e
• (R3)(b) If C has some edges of weight 2, then r(ia , ia+1 )k = e where
k = 4 − wa and wa is the weight of the edge ia − ia−1
Defining the group WΓ with relations as shown above allows them to
prove certain characteristics of the interaction between the relations in this
group for the chordless cycles underlying the diagrams in question. In particular, they prove the following result.
Theorem 2.6. [BM13, Theorem 5.4a] Let Γ be a diagram of finite type and
Γ0 = µk (Γ) the mutation of Γ at vertex k. Then WΓ ∼
= W Γ0 .
The rest of the paper will be devoted to building up analogous relations,
defined in Definition 4.5 to prove a similar result in the case of Artin groups.
For Γ a diagram of finite type, we define the Artin group associated to Γ as
in Section 4. We will then use the group relations presented in this definition
to prove the mutation invariance of AΓ in Section 6.
3. Diagrams of Finite Type
In this section, we shall review the structure of diagrams of finite type,
and how their cycles are affected by mutation. This section is simply a
recap of [BM13, Section 2]. First, in Proposition 3.2, all types of chordless
cycles in diagrams of finite type are classified. Second, in Corollary 3.3 all
possible local pictures between a mutated vertex and two adjacent vertices
6

are drawn. Finally, in Lemma 3.4, all chordless cycles introduced from a
mutation are drawn. These three lemmas will be crucial in proving the main
result Theorem 6.1, as they will allow us to inspect precisely which relations
are added and removed after mutating at a prescribed vertex.
Definition 3.1. A chordless cycle of an unoriented graph G is a connected
subgraph H ⊂ G such that the number of vertices in H is equal to the
number of edges in H, and the edges in H form a single cycle.
Proposition 3.2. [BM13, Proposition 2.1] Let Γ be a diagram of finite type.
Then, a chordless cycle in the unoriented graph of Γ is cyclically oriented in
Γ. Furthermore, the unoriented graph underlying the cycle must either be a
cycle such that all edges have weight 1, a square with two opposite edges of
weight 2 and two opposite edges of weight 1, or a triangle with two edges of
weight 2 and one of weight 1, as pictured in Figure 2.

◦

1

◦

1

1

◦

◦

1

◦
1

◦

◦

2

◦
1

1

◦
2

◦

1

◦

2
◦

2
1

◦

1
◦

◦

Figure 2: Possible chordless cycles in a diagram of finite type, as found in [BM13, Figure
2]

Corollary 3.3. [BM13, Corollary 2.3] Let Γ be a graph of finite type and
suppose there are three vertices, labeled i, j, k with both i, j connected to k.
Then mutation at k on the induced subdiagram appear as in Figure 3, either
from left to right or right to left, up to switching i and j,
Lemma 3.4. [BM13, Lemma 2.5] Let Γ be a diagram of finite type with
Γ0 = µk (Γ), the mutation of Γ at vertex k. In Figure 4 and Figure 5 we list
induced subdiagrams in Γ on the left and the resulting induced subdiagrams
in Γ0 with chordless cycles C 0 on the right, after mutation at k. We draw
7

k
•
(a)

1

k
•
1

◦

1
◦

◦
j

i

1
◦
j

i

k
•
(b)

1

k
•
1

◦

1
◦

◦
j

i

1

i

k
•
(c)

1

2

1
◦

i

2

◦
j

◦
j

i

k
•
2

k
•
1

◦

2
◦

1

◦
j

i

1

2

◦

1
◦

2

◦
j

◦
i

j

k
•
2

2
1

◦

2

i

k
•
2

j

k
•

i

(f)

◦

2

i

k
•
(e)

j

k
•

◦

(d)

◦

1

2

◦

◦
j

1

i

◦
j

Figure 3: Local picture of the mutation of a finite type diagram, as found in [BM13, Figure
4]

8

k=2
•
(a)

1

k=2
•
1

◦

1
◦

1

◦
3

2

1

◦

2
◦

◦

1
◦

1

k=1
•
(e)

◦
4

(f)

◦

1
1

4

1

k=1
•

1

1

2

C0
◦

k=2
•

1

◦

2

◦

◦

1

k=1
•

1

◦
4

◦
3

2

2

3

2

C0
1

◦

2

k=2
•

4

29

◦

1

3

2
◦

1

4

◦

3

k=1
•

3

2

◦

1

1

◦

2
4

2
2

2

2
◦

(g)

◦

2

C0

◦
3

2

2

1

2
◦

1

3

k=2
•
2

1

◦

2

1

k=2
•

◦

2

C0

◦
3

1

3

k=2
•
2

2

◦

2

1

k=2
•

(d)

1

C0

◦
3

1

3

k=2
•

1

(c)

◦

1

1

k=2
•
(b)

1

C0

◦
3
◦

2
C0
1

2
2

◦
3

Figure 4: Induced subdiagrams of Γ and corresponding chordless cycles in Γ0 = µk (Γ), as
found in [BM13, Figure 5]

1
(h)

2

2
◦

1

1
◦

2
◦

2

4
1

1

1

◦
1 h−1 1

1
◦

1

(j) k •

2
◦

1

◦
h

◦

1

◦1

k•

1

1

◦
..
.

◦
h

◦
1 h−1 1

◦1

1

k•

◦2

1

◦

◦
..
.

C0
1

◦

◦2
C0

1
1

..
.
◦
1 h−1 1

3

◦

1
1

◦

1
..
.

◦
h

2

◦

(i) k •
1

k=2
•

C0
◦

3

1

◦

2

◦

1

4
1

1

k=2
•

1

◦

◦
h

◦
1 h−1 1

◦

(k) C is an oriented cycle in Γ not connected to k and C 0 is the corresponding cycle in Γ0 .
1
(l) k •

1

h
◦

1
◦

1

◦2

1
..
.

C0
1

k•

1

h
◦

2
◦

◦
◦
h−1 1 h−2

Figure 5: Induced subdiagrams of Γ and corresponding chordless cycles in Γ0 = µk (Γ), as
found in [BM13, Figure 6]

10

1

C0
1

◦
◦
h−1 1 h−2

◦1

..
.

the diagrams so that C 0 always has a clockwise cycle. Furthermore, in case
(i), we assume C 0 has at least three vertices, while in case (j), we assume
C 0 has at least four vertices.
Every chordless cycle in Γ0 is of one of the types listed in Figure 4 or
Figure 5.
4. The Artin Group of a Diagram
In order to prove our main result, Theorem 6.1, we must first define
the Artin group associated to a finite type diagram. This definition will
be similar to that made in [BM13] at the beginning of Section 3, except
that we shall not require relation (R1), i.e., s2i = e. Since Artin Groups are
very similar to Coxeter groups, with the caveat that the generators are not
involutions, we will be able to use these modified relations to great effect.
4.1. Artin Groups
Notation 4.1. Let
hxi , xj ik =

(
k
(xi xj ) 2 ,
(xi xj )

k−1
2

xi

if k ≡ 0

(mod 2)

if k ≡ 1

(mod 2)

That is, hxi , xj i is just an alternating sequence of xi and xj of length k. We
−1
also write hxi , xj i−k to denote hxi , xj ik
Definition 4.2. [Cha06, Beginning of section 1.2] Let Symn (R) denote the
set of n × n symmetric matrices with entries in R. For M ∈ Symn (Z ∪ ∞)
a symmetric matrix whose entries can take values in the integers or infinity,
we define the associated Artin group in terms of generators and relations by
A = hx1 , . . . , xn |hxi , xj iMi,j = hxj , xi iMi,j for all i, j with Mi,j < ∞i,
Remark 4.3. Each Artin group has an associated Coxeter group defined
by adding in the additional relations s2i = e for all i. An Artin group is
said to be of finite type if its associated Coxeter group is of finite type. To
each Artin group of finite type we can assign to it the same Dynkin diagram
which is assigned to the Coxeter group associated to the Artin group.
One of the most well-known Artin groups is the braid group on n strands,
which was shown to have an Artin group structure in [FN61]. The associated
Coxeter group is the symmetric group Sn .

11

4.2. The Group associated to Diagram
We are now ready to define the Artin group we associate to a diagram
of finite type.
Definition 4.4. Let (i0 , . . . , id−1 ) be an ordered tuple such that the subgraph of Γ on the vertices i0 , . . . , id−1 is a chordless cycle, with edges of
nonzero weight from ik to ik+1 , where subscripts are taken (mod d). Call
such an ordered tuple a chordless cycle tuple. Then, denote
−1
−1
p(ia , ia+1 ) = s−1
ia+1 sia+2 . . . sia−2 sia−1 sia−2 sia−3 . . . sia+1 .

Additionally, let
t(ia , ia+1 ) = [sia , p(ia , ia+1 )]
where [a, b] = aba−1 b−1 is the commutator.
Definition 4.5. The associated Artin group to a diagram Γ of finite type,
denoted AΓ , is generated by si , where there is one si for each vertex i in Γ.
These generators are subject to the following relations
(T2) With mij as defined in Definition 2.3, for all i 6= j, we add the relations
hsi , sj imij = hsj , si imij .
(T3) Let (i0 , i1 , . . . , id−1 ) be a chordless cycle tuple, as defined in Definition
4.4. If additionally one of the following two conditions hold,
1. All edges in the chordless cycle are of weight 1 or 2 and the edge
id−1 → i0 has weight 2,
2. All edges in the chordless cycle have weight 1,
then, we include the relation t(i0 , i1 ) = e. That is, si0 and p(i0 , i1 )
commute.
Remark 4.6. In the above definition, the chordless cycle tuple is ordered,
and so we may have other relations corresponding to chordless cycle tuples
which are cyclic reorderings the chordless cycle tuple (i0 , . . . , id−1 ). However,
we shall see in Section 5 that many of these relations are redundant.
Remark 4.7. We purposely include relations (T 2), (T 3) but not (T 1) in
order to make our relation labeling analogous to that of [BM13] at the
beginning of Section 3. Note that if we add the additional relation (R1) as
defined at the beginning of Section 3 of [BM13] (namely, if we add s2i = e
for all vertices i in Γ), then we will precisely obtain the group WΓ as defined
at the beginning of Section 3 in [BM13].
12

Remark 4.8. Throughout the remainder of the paper, we shall frequently
discuss relations on one diagram of finite type, Γ, and another diagram of
finite type Γ0 . In order to distinguish the relations in these two groups, we
shall refer to the relations on Γ as (T 2), (T 3) and the relations on Γ0 as
(T 20 ), (T 30 ).
Example 4.9. The relations (T 2), (T 3) in that Γ is a square with all edges
of weight 1 are as follows:
1
Γ=

1

◦

1

1
◦

◦

1

4
(T2)

2

◦

3

• hs1 , s2 i3 = hs2 , s1 i3 , i.e., s1 s2 s1 = s2 s1 s2
• hs2 , s3 i3 = hs3 , s2 i3 , i.e., s2 s3 s2 = s3 s2 s3
• hs3 , s4 i3 = hs4 , s3 i3 , i.e., s3 s4 s3 = s4 s3 s4
• hs4 , s1 i3 = hs1 , s4 i3 , i.e., s4 s1 s4 = s1 s4 s1

(T3)

−1 −1 −1 −1
−1
• s1 s−1
2 s3 s4 s3 s2 s1 s2 s3 s4 s3 s2 = e
−1 −1 −1 −1
−1
• s2 s−1
3 s4 s1 s4 s3 s2 s3 s4 s1 s4 s3 = e
−1 −1 −1 −1
−1
• s3 s−1
4 s1 s2 s1 s4 s3 s4 s1 s2 s1 s4 = e
−1 −1 −1 −1
−1
• s4 s−1
1 s2 s3 s2 s1 s4 s1 s2 s3 s2 s1 = e

Example 4.10. The relations (T 2), (T 3) in that Γ is a triangle with two
edges of weight 2 and one of weight 1 are as follows:
1
◦
Γ=

2
◦

1

3
(T2)

2
◦
2

• hs1 , s2 i4 = hs2 , s1 i4 , i.e., s1 s2 s1 s2 = s2 s1 s2 s1
• hs2 , s3 i3 = hs3 , s2 i3 , i.e., s2 s3 s2 = s3 s2 s3
• hs3 , s4 i4 = hs4 , s3 i4 , i.e., s3 s4 s3 s4 = s4 s3 s4 s3

(T3)

−1
−1 −1
• s1 s−1
2 s3 s2 s1 s2 s3 s2 = e

13

−1
−1 −1
• s2 s−1
3 s1 s3 s2 s3 s1 s3 = e

Remark 4.11. Note that if Γ is the graph associated to a Dynkin diagram,
then WΓ as we have defined it is precisely the Artin group corresponding to
that Dynkin diagram. This occurs because, in this case, we have no cycles
in Γ, and so we only have relations of the form (T 2), which define the Artin
group.
5. Symmetry among the (R3) Relations
Given the relations (T2), many of the relations in (T3) become redundant. For example,
Lemma 5.1. Let Γ be a diagram of finite type which contains a chordless
cycle C:
i0
i1
···
id−1
i0
so that all edges have weight 1. Then if W is a group generated by s1 , . . . , sn
satisfying the relations (T2) and t(ia , ia+1 ) = e for some a ∈ {1, . . . , d}, all
of the relations in (T3) hold for C.
In the proof of this Lemma and throughout the rest of the paper, we
will frequently employ the following relations, which follow from the (T2)
relations.
Lemma 5.2. Let Γ be a diagram of finite type, and let AΓ be generated by
s1 , . . . , sn . Then we have that
−1
(a) si sj s−1
i = sj si sj if there is an arrow of weight 1 from i to j in Γ

−1
−1 −1
(b) si sj s−1
i sj = sj si sj si if there is an arrow of weight 2 from i to j in
Γ

Proof of 5.2. The relation in (a) follows immediately from the (T2) relation
sj si sj = si sj si by left multiplying both sides by s−1
j and right multiplying
−1
both sides by si . Similarly, the relation in (b) arises from the (T2) rela−1
tion sj si sj si = si sj si sj by left multiplying both sides by s−1
and right
i sj
−1 −1
multiplying both sides by sj si

14

Proof of 5.1. It suffices to prove that the relation t(0, 1) = e implies that
t(d-1, 0) = e, as the other relations will follow by induction. So suppose AΓ
satisfies the relation t(0, 1) = e. Then we have
−1
−1
sd−1 p(d − 1, 0) = sd−1 s−1
0 s1 . . . sd−3 sd−2 sd−3 . . . s1 s0
−1 −1
−1
−1
= s−1
0 s0 sd−1 s0 s1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 sd−1 s0
−1
−1
−1
−1
= s−1
0 sd−1 s0 sd−1 s1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 sd−1 s0

=
=
=
=
=
=
=
=

−1
−1
−1
−1
s−1
0 sd−1 s0 s1 . . . sd−3 sd−1 sd−2 sd−1 sd−3 . . . s1 sd−1 s0
−1
−1
−1 −1
s−1
0 sd−1 (s0 s1 . . . sd−3 sd−2 sd−1 sd−2 sd−3 . . . s1 )sd−1 s0
−1
−1
−1 −1
s−1
0 sd−1 (s1 . . . sd−3 sd−2 sd−1 sd−2 sd−3 . . . s0 )sd−1 s0
−1
−1
−1
−1
s−1
0 sd−1 (s1 . . . sd−3 sd−1 sd−2 sd−1 sd−3 . . . s0 )sd−1 s0
−1
−1
−1
−1
s−1
0 (sd−1 sd−1 )s1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 s0 sd−1 s0
−1
−1
−1
s−1
0 s1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 s0 sd−1 s0
−1
−1
−1
s−1
0 s1 . . . sd−3 sd−2 sd−3 . . . s1 s0 sd−1 s0 s0
−1
−1
s−1
0 s1 . . . sd−3 sd−2 sd−3 . . . s1 s0 sd−1 = p(d − 1, 0)sd−1

by (T2)
by (T2)
by t(0, 1) = e

as required. Note that line 3 is equal to 4 and line 7 is equal to line 8 since the
cycle is chordless, meaning that sd−1 commutes with every element except
s0 and sd−2 .
Furthermore, we obtain similar results for cycles containing edges of
weight 2.
Lemma 5.3. Let Γ be a diagram of finite type containing the following
3-cycle:
k◦
2

2

and let A be the group with generators s1 , . . . , sn de-

◦i
1
fined by Γ. Then the relations t(i, j) = e and t(k, i) = e are equivalent.
j◦

15

by (T2)

by (T2)

Proof. The lemma follows from the fact that
−1
−1 −1
s−1
k sj (si p(i, j)si p(i, j) )sj sk
−1
−1 −1 −1
−1
= s−1
k sj (si sj sk sj si sj sk sj )sj sk
−1
−1 −1
= s−1
k sj si sj sk sj si sj
−1
−1 −1
= s−1
k si sj si sk si sj si
−1
= s−1
k p(k, i)sk p(k, i)

In the setting of the previous lemma, we also obtain the following relation, which will play an important role in later proofs.
Lemma 5.4. Suppose Γ contains a 3-cycle with edges of weight 2, labeled
as in 5.3, and suppose that AΓ is generated by s1 , . . . sn . Then we have that
−1 −1 −1
−1 −1
−1
sj p(j, k)sj p(j, k)−1 s−1
:= sj s−1
j p(j, k)
k si sk sj sk si sk sj sk si sk = e.

Proof. We show that
−1 −1 −1
−1
−1
s−1
j sk si sk sj sk si sk sj sk si sk = e.

The result then follows by inverting the relation and conjugating by sj . In
the following computation, we will underline the terms being manipulated
in each line for emphasis.

16

−1
−1
−1 −1 −1
−1 −1 −1
sk sj sk (s−1
j sk si sk sj sk si sk sj sk si sk )sk sj sk
−1
−1 −1 −1
−1 −1 −1
−1
= sk sj sk s−1
j sk si sk sj sk si sk sj sk si sk sk sj sk
−1
−1
−1 −1 −1 −1 −1
= sk s−1
k sj sk sj si sk sj sk si sk sj sk si sj sk
−1
−1 −1 −1 −1 −1
= si s−1
j sk sj sk sj sk si sk sj sk si sj sk
−1 −1 −1 −1 −1
−1
= si sk sj sk s−1
j sj sk si sk sj sk si sj sk
−1 −1 −1
= si esk sj si sk sj−1 s−1
k si sj sk
−1 −1 −1 −1 −1
= si sj s−1
j sk sj si sk sj sk si sj sk
−1 −1 −1 −1 −1
= si sj si s−1
j sk sj sk sj sk si sj sk
−1 −1 −1 −1 −1
= sj si sj s−1
j sk sj sk sj sk si sj sk
−1
−1 −1 −1
= sj si sk s−1
k sj sk sj si sj sk
−1
−1 −1
= sj si s−1
i sj sk sj sj sk
−1
= sj s−1
j sk sk

=e

Lemma 5.5. Let Γ be a diagram of finite type containing the following 41
i◦
◦j
cycle:

2

2 and let A be the group with generators s1 , . . . , sn

◦k
1
defined by Γ. Then the relations t(i, j) = e and t(k, l) = e are equivalent.
l◦

Proof. We have that
−1
−1
−1 −1
s−1
k sl sj (si p(i, j)si p(i, j) )sj sl sk
−1
−1 −1
−1 −1 −1 −1
−1
= s−1
k sl sj (si sj sk sl sk sj si sj sk sl sk sj )sj sl sk
−1
−1
−1
−1 −1 −1 −1
= s−1
k sl (sj si sj )(sk sl sk )(sj si sj )sk sl sk sl sk
−1 −1
−1 −1 −1
−1 −1
= s−1
k sl si sj si sl sk sl si sj si sk (sl sk sl )sk
−1 −1
−1 −1 −1
= s−1
k sl si sj si sl sk sl si sj si sl
−1
= s−1
k p(k, l)sk p(k, l)

17

Finally, we conclude the section by establishing a relationship between
the groups defined by Γ and Γop , the diagram obtained by reversing all
arrows in Γ.
Lemma 5.6. Let AΓ be generated by s1 , . . . , sn , and let AΓop be generated
by r1 , . . . , rn . Then the map
∆ : si → ri−1
defines an isomorphism between AΓ and AΓop .
Proof. It suffices to show that the map is well-defined, or that the elements
−1
s−1
1 , . . . , sn satisfy the relations (T2) and (T3) of AΓop . One can see that
the inverse elements satisfy (T2) in AΓop by taking the inverse of both sides
of the corresponding relation in AΓ . To see that the elements satisfy (T3)
in AΓop , note that for a chordless cycle in Γ with all weights equal to one,
we have
−1
−1
−1
s0 s−1
1 . . . sd−2 sd−1 sd−2 . . . s1 = s1 . . . sd−2 sd−1 sd−2 . . . s1 s0

by the relation t(0,1) = e in (T3) in AΓ . But then applying relations from
(T2), we have that
−1
−1
−1
−1
s0 s−1
1 . . . sd−3 sd−1 sd−2 sd−1 sd−3 . . . s1 = s1 . . . sd−1 sd−2 sd−1 . . . s1 s0 ,

and since the cycle is chordless, we then have
−1
−1
−1
−1
−1
s0 sd−1 s−1
1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 = sd−1 s1 . . . sd−3 sd−2 sd−3 . . . s1 sd−1 s0 .

Repeating this process, we find that
−1 −1
−1
−1 −1
s0 sd−1 sd−2 . . . s2 s1 s−1
2 . . . sd−2 sd−1 = sd−1 sd−2 . . . s2 s1 s2 . . . sd−2 sd−1 s0 .
−1
But this occurs if and only if s−1
1 , . . . , sn satisfies the relation t(0, d-1) = e
in AΓop .
For a triangle labeled as in 5.3, by the relation t(k, i) = e we have
−1
−1
sk s−1
i sj si sk = si sj si .

Hence
−1
−1
sk sj si s−1
j sk = sj si sj .

18

−1 −1
But as before, this can occur if and only if s−1
i , sj , sk satisfy the relation
t(k, j) = e in AΓop .
Finally, given a square labeled as in 5.5 and the relations t(1, 2) = e
and t(3, 4) = e, we have
−1
sj si sl sk s−1
l si
−1 −1
= si s−1
i sj si sl sk sl si
−1
−1
= si sj si s−1
j sk sl sk si
−1
−1 −1
= si sj (si s−1
j sk sl sk sj )sj si
−1
−1 −1
= si sj s−1
j (sk sl sk )sj (si sj si )
−1 −1
= si sl sk s−1
l sj sj si sj
−1
= si sl sk s−1
l si sj
−1
But this relation holds if and only if s−1
satisfy t(j, i) = e in AΓop .
i , . . . , sl
Therefore, we are done.

6. Main Result
In Section 6.1 we prove our main result:
Theorem 6.1. Let Γ be a diagram of finite type, and let Γ0 = µk (Γ) be the
mutation of Γ at vertex k. Then AΓ ∼
= A Γ0
The structure of the proof will be analogous to the structure of the proof
of Theorem A in [BM13], but with most of the details changed to account
for the fact that the (R1) relations are not included. In particular, many of
the computations used to prove the supporting lemmas and propositions of
Theorem A rely heavily on the (R1) relations and therefore do not apply to
the supporting lemmas and propositions of Theorem 6.1.
Throughout the section we will fix a diagram of finite type Γ, a vertex k
of Γ, and write Γ0 = µk (Γ). We will write si , ri , qi , and ui for the generators
corresponding to vertex i of AΓ , AΓ0 , AΓop , and A(Γ0 )op , respectively. Note
that the ui are generators A(Γop )0 as well, since (Γ0 )op = (Γop )0 . In the proof
of Theorem 6.1 we will use Lemma 5.6 along with the following proposition,
which we prove in Section 6.2.
Proposition 6.2. The map ϕ : AΓ0 → AΓ defined by
(
sk si s−1
if there is a (possibly weighted) arrow i → k in Γ
k
ϕ(ri ) =
si
otherwise
is a group homomorphism.
19

6.1. Proof of Main Result
Proof of Theorem 6.1. By Proposition 6.2 ϕ : AΓ0 → AΓ is a group homomorphism and ϕop : AΓop → A(Γop )0 defined by
(
uk ui u−1
k
ϕop (qi ) =
ui

if there is a (possibly weighted) arrow i → k in (Γop )0
otherwise

is a group homomorphism as well. By Lemma 5.6, there exist two well−1
0
op
defined homomorphisms ∆ : AΓ → Aop
Γ defined by ∆(si ) = qi and ∆ : A(Γ0 ) →
−1
0
AΓ0 defined by ∆ (ui ) = ri . We then have a homomorphism
ψ = ∆0 ◦ ϕop ◦ ∆ : A(Γ) → A(Γop ) → A((Γop )0 ) → A(Γ0 )
Suppose that there is an arrow i → k in Γ. Then there will be an arrow
k → i in Γop and hence an arrow i → k in (Γop )0 , so we have that
−1 −1
0
0 −1
ψ◦ϕ(ri ) = ∆0 (ϕop (∆(ϕ(ri )))) = ∆0 (ϕop (∆(sk si s−1
k ))) = ∆ (ϕop (qk qi qk )) = ∆ (ui ) = ri

Similarly if there is an arrow k → i or no arrow between i and k in Γ
then there will be an arrow k → i or no arrow between i and k in (Γop )0 ,
respectively. In each of these cases we have that
ψ◦ϕ(ri ) = ∆0 (ϕop (∆(ϕ(ri )))) = ∆0 (ϕop (∆(si ))) = ∆0 (ϕop (qi−1 )) = ∆0 (u−1
i ) = ri
In other words, starting at any node in the square below and following
the maps around gives the identity map.
AΓop ◦

ϕop

Γ0

∆
AΓ ◦

◦A(Γ0 )op

ϕ

◦AΓ0

Thus ψ ◦ ϕ is the identity map on AΓ0 . By a similar argument ϕ ◦ ψ is
the identity map on AΓ , and hence AΓ ∼
= A Γ0 .

20

6.2. Proof of Proposition 6.2
We prove Proposition 6.2 by showing that that the elements ϕ(ri ) ∈ AΓ
satisfy the (T 20 ) and (T 30 ) relations in AΓ0 . The proof that the ϕ(ri ) satisfy
these relations is divided among Lemmas 6.3, 6.4, and 6.5. Throughout the
proofs we write ti = ϕ(ri ) and m0ij for the weight of the edge between i and
j in Γ0 .
Lemma 6.3. Let i, j be distinct vertices of Γ.
0

0

(a) If i = k or j = k, then hti tj imij = htj ti imij .
0

0

(b) If at most one of i, j is connected to k in Γ, then hti tj imij = htj ti imij .
Proof. For case (a), suppose without loss of generality that i = k. Note that
m0ij = mij . The only nontrivial case is when there is an arrow j → k = i.
Since i and j are connected in this case, mij is one of 3, 4, or 6.
Case mij = 3. Here hsj si i3 = hsi sj i3 , so si sj = sj si sj s−1
and we have
i
−1
3
hti tj i3 = ti tj ti = si si sj s−1
i si = si si sj = si sj si sj si = tj ti tj = htj ti i

Case mij = 4. Here hsi sj i4 = hsj si i4 , so si si sj si sj s−1
i = si sj si sj and therefore
−1
−1
−1
−1
4
hti tj i4 = si si sj s−1
i si si sj si = si si sj si sj si = si sj si sj = si sj si si si sj si si = htj ti i

Case mij = 6. Here hsi sj i6 = hsj si i6 , so si si sj si sj si sj s−1
i = si sj si sj si sj . As
in the previous case, we add and remove pairs si s−1
as
necessary, giving
i
−1
−1
−1
−1
−1
6
hti tj i6 = si si sj s−1
i si si sj si si si sj si = si sj si si si sj si si si sj si si = hti tj i

For case (b), the only nontrivial case is when there is an arrow i → k or
j → k. Without loss of generality, suppose there is an arrow i → k. Since j
is not connected to k, we know that sj sk = sk sj .
Case mij = 2. Here si sj = sj si , so sj commutes with both si and sk and we
have that
−1
ti tj = sk si s−1
k sj = sj sk si sk = tj ti

21

Case mij = 3. Here hsi sj i3 = hsj si i3 , so we have that
−1
−1
−1
−1
3
hti tj i3 = sk si s−1
k sj sk si sk = sk si sj si sk = sk sj si sj sk = sj sk si sk sj = htj ti i

Case mij = 4. Here hsi sj i4 = hsj si i4 , so we have that
−1
−1
−1
−1
−1
4
hti tj i4 = sk si s−1
k sj sk si sk sj = sk si sj si sj sk = sk sj si sj si sk = sj sk si sk sj sk si sk = htj ti i

Case mij = 6. Here hsi sj i6 = hsj si i6 , so we have that
−1
−1
−1
−1
6
hti tj i6 = sk si s−1
k sj sk si sk sj sk si sk sj = sk si sj si sj si sj sk = sk sj si sj si sj si sk = hti tj i

Lemma 6.4. Let i, j be distinct vertices of Γ such that i and j are connected.
0
0
Then hti tj imij = htj ti imij .
Proof. The possibilities for the subdiagram induced by i, j, and k are enumerated in Figure 3. We show that ti and tj satisfy the (T 20 ) relations by
checking each case. Within each case, subcase (i) is when the subdiagram
of Γ is the diagram on the left in Figure 3, and subcase (ii) is when the
subdiagram of Γ is the diagram on the right in Figure 3.
Throughout the proof we will make frequent use of the fact that if m
and n are vertices of Γ, then
−1
−1 −1
−1 −1
−1 −1 −1
−1 −1 −1
sm sn sm = sn sm sn ⇔ sm sn s−1
m = sn sm sn ⇔ sm sn sm = sn sm sn ⇔ sm sn sm = sn sm sn

When helpful, we underline the sections of an expression that are about
to be manipulated. We also frequently combine two applications of AΓ
relations when one manipulation is simply commuting pairs of variables.
a)

−1
−1
−1
2
i) We have hti tj i2 = sk si s−1
k sk sj sk = sk si sj sk = sk sj si sk = htj ti i .

ii) We have hti tj i2 = si sj = sj si = htj ti i2 .

22

b)

i) We have
−1
hti tj i3 = sk si s−1
k sj sk si sk
−1
= sk si sj sk s−1
j si sk
−1
= sk sj si sk si s−1
j sk
−1
= sk sj sk si sk s−1
j sk
−1
= sj sk sj si s−1
j sk sj

= sj sk si s−1
k sj
= htj ti i3
−1
−1
−1
ii) We have hti tj i2 = si sk sj s−1
k = si sj sk sj = sj sk sj si = sk sj sk si =

htj ti i2
c)

−1
−1
−1
2
i) We have hti tj i2 = sk si s−1
k sk sj sk = sk si sj sk = sk sj si sk = htj ti i

ii) We have hti tj i2 = si sj = sj si = htj ti i2
d)

i) We have
−1 −1 −1
hti tj i4 htj ti i−4 = ti tj ti tj t−1
i tj ti tj
−1
−1 −1 −1
−1 −1 −1
= sk si s−1
k sj sk si sk sj sk si sk sj sk si sk sj
−1 −1
−1 −1 −1 −1 −1
= sk si s−1
k sj sk si sj sk sj si sj sk sj si sk sj
−1 −1 −1 −1 −1 −1
= sk si s−1
k sk sj sk si sk si sk sj si sk sj
−1 −1 −1 −1 −1
= sk sj si sk si sk s−1
i sk si sk sj sk
−1
= sk sj hsi sj i4 hsj si i−4 s−1
j sk

=e
−1
−1
−1
ii) We have hti tj i2 = si sk sj s−1
k = si sj sk sj = sj sk sj si = sk sj sk si =
2
htj ti i

23

e)

i) We have
−1 −1 −1
hti tj i4 htj ti i−4 = ti tj ti tj t−1
i tj ti tj
−1
−1 −1 −1
−1 −1 −1
= sk si s−1
k sj sk si sk sj sk si sk sj sk si sk sj
−1 −1
−1 −1 −1
−1
−1
= s−1
i sk si sj si sk si sj si sk si sj si sk si sj
−1 −1 −1 −1
= s−1
i sk sj sk sj sk sj sk sj si
4
−4
= s−1
i htk tj i htj tk i si

=e
−1 −1
−1 −1
−1
ii) Since sj s−1
k si sk = sk si sk sj , we have that sj sk si sk = sk si sk sj ,
hence

−1
hti tj i2 htj ti i−2 = ti tj t−1
i tj
−1
−1 −1
= si sk sj s−1
k si sk sj sk
−1
−1 −1
= si sk s−1
k si sk sj sj sk

=e

f)

i) We have that
−1 −1 −1
−1
3
−3
s−1
k hti tj i htj ti i sk = sk ti tj ti tj ti tj sk
−1 −1
−1 −1 −1
= si s−1
k sj sk si sk sj sk si sk sj sk

=e
Where the second equality follows from Lemma 5.4. It follows that
hti tj i4 = htj ti i4 .
ii) This follows from part (i) by symmetry.

Lemma 6.5. The elements ti satisfy the (T 30 ) relations in Γ0 .
Proof. We know that every chordless cycle in Γ0 arises from a subdiagram
of Γ in the form of one of the cases of Lemma 3.4, so we simply need to
check that a cycle relation holds in each case. We follow the labeling of the
vertices used in Lemma 3.4. We denote by t0 (m, n) the expression given by
24

replacing the generators ri in the corresponding expression t(m, n) in AΓ0
with ti . When helpful we note uses of (T 3) relations in AΓ by referencing
the particular cycle relation used next to the manipulation.
−1
−1
−1
a) We have t1 t−1
2 t3 t2 = s1 s2 s2 s3 s2 s2 = s1 s3 = s3 s1 = t2 t3 t2 t1 , hence
0
t (1, 2) = e.
−1
−1
−1
b) We have t1 t−1
2 t3 t2 = s1 s2 s2 s3 s2 s2 = s1 s3 = s3 s1 = t2 t3 t2 t1 , hence
0
t (1, 2) = e.
−1
−1
−1
0
c) We have t1 t−1
2 t3 t2 = s1 s2 s2 s3 s2 s2 = s3 s1 = t2 t3 t2 t1 , hence t (1, 2) =
e.

d) We have
−1 −1 −1
−1
−1
0
s−1
2 t (3, 1)s2 = s2 t3 t1 t2 t1 t3 t1 s2 t1 s2
−1
−1 −1 −1 −1
= s3 s−1
2 s1 s2 s1 s2 s3 s2 s1 s2 s1 s2
−1
−1 −1
= s3 s1 s2 s−1
1 s3 s1 s2 s1
−1 −1
−1
= s3 s1 s−1
1 s3 s1 s2 s2 s1

by t(2, 1)

=e
Hence t0 (3, 1) = e.
e) We have
−1
−1 −1 −1
−1
−1 −1
−1
t1 t−1
2 t3 t4 t3 t2 = (s1 s2 s1 s1 s2 s1 )s1 s1 s2 s1 s3 s4 s3 s1 s2 s1
−1 −1 −1
−1
−1
= s1 s−1
2 s1 s1 s2 s1 s2 s1 s3 s4 s3 s1 s2 s1
−1
−1
−1
= s1 s−1
2 s1 s2 s3 s4 s3 s1 s2 s1
−1 −1
−1
= s1 s−1
2 s1 s3 s4 s3 s2 s1 s2 s1
−1 −1
−1
= s1 s−1
2 s1 s3 s4 s3 s1 s2 s1 s1
−1
= t−1
2 t3 t4 t3 t2 t1

Hence t0 (1, 2) = e.

25

by t(2, 3)

f) We have
−1
−1
−1 −1
−1
t1 t−1
2 t3 t4 t3 t2 = s1 s2 s2 s3 s2 s4 s2 s3 s2 s2

= s1 s−1
3 s4 s3
= s−1
3 s4 s3 s1
=
=
=

by t(1, 3)

−1
−1 −1
−1
(s−1
2 s2 )s3 (s2 s2 )s4 (s2 s2 )s3 (s2 s2 s1
−1 −1
−1
s−1
2 s2 s3 s2 s4 s2 s3 s2 s2 s1
−1
t−1
2 t3 t4 t3 t2 t1

Hence t0 (1, 2) = e.
g) We have
−1 −1 −1
t0 (3, 4) = t3 t−1
4 t2 t4 t3 t4 t2 t4
−1
−1 −1
−1 −1 −1
= s3 s1 s4−1 s−1
1 s2 s1 s4 s1 s3 s1 s4 s1 s2 s1 s4 s1
−1 −1 −1 −1
−1
= s1 s3 s4−1 s−1
1 s2 s1 s4 s3 s4 s1 s2 s1 s4 s1

= s1 t(3, 4)s−1
1
= s1 s−1
1
=e

h) We have
−1
−1
t3 t−1
4 t1 t4 = s3 s4 s2 s1 s2 s4
−1
= s3 s−1
4 s1 s2 s1 s4
−1
= s−1
4 s1 s2 s1 s4 s3
−1
= s−1
4 s2 s1 s2 s4 s3

= t−1
4 t1 t4 t3

Hence t0 (3, 4) = e.

26

byt(3, 4)

i) We have
t0 (1, 2)
−1
−1
−1 −1
−1 −1
= t1 t−1
2 t3 · · · th−1 th th−1 · · · t2 t1 t2 · · · th−1 th th−1 · · · t2
−1
−1
−1
−1 −1
−1
−1 −1
= s1 s−1
2 s3 · · · sh−1 sk sh sk sh−1 · · · s2 s1 s2 · · · sh−1 sk sh sk sh−1 · · · s2
−1
−1 −1
−1 −1
−1 −1 −1
= s1 s−1
2 s3 · · · sh−1 sh sk sh sh−1 · · · s2 s1 s2 · · · sh−1 sh sk sh sh−1 · · · s2

= t(1, 2)
=e

j) Here sk commutes with si for all i 6= 1, h, so we have
t0 (h, 1)
−1 −1
−1
−1 −1 −1 −1
−1 −1
= th t−1
k t1 t2 · · · th−2 th−1 th−2 · · · t1 tk th tk t1 t2 · · · th−2 th−1 th−2 · · · t2 t1 tk
−1
−1 −1 −1
−1
−1 −1
−1
= sh s−1
k sk s1 sk s2 · · · sh−2 sh−1 sh−2 · · · sk s1 sk sk sh sk sk s1
−1
−1 −1
−1
s−1
k s2 · · · sh−2 sh−1 sh−2 · · · s2 sk s1 sk sk
−1 −1
−1 −1 −1
−1
−1
= sh s−1
1 s2 · · · sh−2 sh−1 sh−2 · · · s2 s1 sh s1 s2 · · · sh−2 sh−1 sh−2 · · · s2 s1

= t(h, 1)
=e

k) Here ti = si for all vertices i in C 0 , so the case is trivial.
l) If the edge between k and h points towards h, then ti = si for all i ∈ C 0
and the case is trivial. If the edge points towards k then th = sk sh s−1
k
and sk commutes with si for all vertices i 6= h in C 0 , so we have
−1
−1
−1
−1
t1 t−1
2 · · · th−1 th th−1 · · · t2 = s1 s2 · · · sh−1 sk sh sk sh−1 · · · s2
−1
−1
= sk s1 s−1
2 · · · sh−1 sh sh−1 · · · s2 sk
−1
−1
= sk s−1
by t(1, 2)
2 · · · sh−1 sh sh−1 · · · s2 s1 sk
−1
−1
= s−1
2 · · · sh−1 sk sh sk sh−1 · · · s2 s1
−1
= t−1
2 · · · th−1 th th−1 · · · t2 t1

Hence t0 (1, 2) = e.
27

Proof of Proposition 6.2. Lemma 6.3 and Lemma 6.4 show that the elements
ti satisfy the (T 20 ) relations for AΓ0 . Lemma 6.5 shows that they satisfy the
(T 30 ) relations. Since these are all of the relations defining AΓ0 , it follows
that ϕ defines a group homomorphism AΓ0 → AΓ .
7. Extension to Affine Type Diagrams
It is natural to ask whether one can obtain similar results for diagrams
that are not of finite type, particularly those of affine type (i.e. mutation
equivalent to an affine Dynkin diagram). In [FT13], the authors associated
the following group to a diagram of affine type.
Let Γ be a diagram of affine type with n + 1 vertices. Then we define
WΓ to be the diagram with generators s1 , . . . , sn and satisfying the following
relations:
(R1) s2i = e for all i = 1, . . . , n
(R2) (si sj )mij = e where


2
if there is





if there is
3
mij = 4
if there is



6
if there is




∞ otherwise

no
an
an
an

arrow
arrow
arrow
arrow

between i and j in Γ
of weight 1 between i and j in Γ
of weight 2 between i and j in Γ
of weight 3 between i and j in Γ

(R3) For every chordless oriented cycle:
wi

wid−2

wi

wid−1

0
1
i0 −→
i1 −→
· · · −→ id−1 −→ i0 ,

define for l ∈ {0, . . . , d − 1},
t(l) = (

l+d−2
Y

√
√
wij − wil+d−1 )2 .

j=l

Then take the relation (sil p(il , il+1 ))m(l) = e where


2 if t(l) = 0



3 if t(l) = 1
m(l) =

4 if t(l) = 2



6 if t(l) = 3
28

(R4) For each subdiagram of Γ of the form shown in the first column of
Table 1, we add the relation(s) listed in the second column.
Given this definition of WΓ , which generalizes the definition of WΓ found
in [BM13], one obtains the following result.
Theorem 7.1. [FT13, Theorem 4.6] Let W be an affine Weyl group and let
Γ be a diagram mutation equivalent to an orientation of a Dynkin diagram
of the same type as W. Then W is isomorphic to WΓ .
In seeking an analogous result for Artin groups, we generalize AΓ as
follows:
Definition 7.2. Let Γ be a diagram of affine type with n + 1 vertices. Then
we define AΓ to be the diagram with generators s1 , . . . , sn and satisfying the
following relations:
(T2) hsi , sj imij = hsj , si imij where


2
if there is no arrow





if there is an arrow
3
mij = 4
if there is an arrow



6
if there is an arrow




∞ otherwise

between i and j in Γ
of weight 1 between i and j in Γ
of weight 2 between i and j in Γ
of weight 3 between i and j in Γ

(T3) For every chordless oriented cycle:
wi

wid−2

wi

wid−1

0
1
i0 −→
i1 −→
· · · −→ id−1 −→ i0 ,

define for l ∈ {0, . . . , d − 1},
t(l) = (

l+d−2
Y

√
√
wij − wil+d−1 )2 .

j=l

Then take the relation hsil p(il , il+1 )im(l) = hp(il , il+1 )sil im(l) where


2 if t(l) = 0



3 if t(l) = 1
m(l) =

4 if t(l) = 2



6 if t(l) = 3
29

(T4) For each subdiagram of Γ of the form shown in the first column of
Table 2, we add the relation(s) listed in the second column.
Conjecture 7.3. Let Γ be a diagram of affine type, and let Γ0 be the mutation of Γ at a node k. Then AΓ ∼
= A Γ0 .

8. Acknowledgements
This work was supported by the RTG grant NSF/DMS-1148634. We
would also like to thank the University of Minnesota School of Mathematics
for hosting us. We are greatly appreciative of Gregg Musiker’s mentoring,
support, and guidance on this problem. We thank Emily Gunawan for her
useful comments on our work. We also thank Vic Reiner for motivating this
problem.

References
[BM13] Michael Barot and Robert J. Marsh. Reflection group presentations
arising from cluster algebras. arXiv: 1112.2300v2, 2013.
[Cha06] Ruth Charney. An introduction to right-angled artin groups.
arXiv:math/0610668, 2006.
[FN61] R. Fox and L. Neuwirth. The braid groups. Mathematica Scandinavica, pages 119–126, 1961.
[FT13] A. Felikson and P. Tumarkin. Coxeter groups and their quotients
arising from cluster algebras. arXiv:1307.0672v2, 2013.
[FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras I: Foundations. J. Amer. Math. Soc., 15:497–529, 2002.
[FZ03] Sergey Fomin and Andrei Zelevinsky. Cluster algebras II: Finite
type classification. Invent. Math., 154:63–121, 2003.
[GM14] Joseph Grant and Robert J. Marsh. Braid groups and quiver mutation. arXiv:1408.5267, 2014.
[Qiu14] Y. Qiu. On the spherical twists on 3-calabi-yau categories from
marked surfaces. arXiv:1407.0806, 2014.

30

Subdiagram

(R4) Relation
(s1 s2 s3 s4 s3 s2 )2 = e

2

4

1

3

4

(s1 s2 s3 s2 s1 s4 s5 . . . sn sn+1 sn . . . s5 s4 )2 = e
1

n+1

3

n

4
2
5

(s2 s3 s4 s1 s4 s3 )2 = e and (s2 s1 s4 s3 s4 s1 )2 =
e

2
2

2
4

1

3

4

(sn+1 s1 sn+1 s2 s3 . . . sn−1 sn sn−1 . . . s3 s2 )2 =
e

n+1
2

2

n

1

n-1

2

3

(s2 s1 s2 s1 s2 s3 )2 = e and (s2 s3 s2 s3 s2 s1 )2 =
e

2

3

3

1

3
4

Table 1: (R4) Relations

31

Subdiagram

(T4) Relation
−1
−1
−1
s2 s1 s−1
2 s4 s3 s4 = s4 s3 s4 s2 s1 s2

2

4

1

3

4

−1
−1
−1
s2 s−1
3 s4 . . . sn s1 sn+1 s1 sn . . . s4 s3
−1 −1
−1
= s3 s4 . . . sn s1 sn+1 s−1
1 sn . . . s4 s3 s2

1

n+1

3

n

4
2
5

−1
−1
−1
s2 s−1
3 s4 s1 s4 s3 = s3 s4 s1 s4 s3 s2 and
−1
−1
−1
−1
s2 s1 s4 s3 s4 s1 = s1 s4 s3 s4 s1 s2

2
2

2
4

1

3

4

−1
−1
s1 s−1
2 . . . sn−1 sn+1 sn sn+1 sn−1 . . . s2
−1
−1
s−1
2 . . . sn−1 sn+1 sn sn+1 sn−1 . . . s2 s1

n+1
2

=

2

n

1

n-1

2

3

−1
−1
−1
s2 s−1
1 s2 s3 s2 s1 = s1 s2 s3 s2 s1 s2 and
−1
−1 −1
s1 s2 s3 s2 s−1
3 s2 = s2 s3 s2 s3 s2 s1

2

3

3

1

3
4

Table 2: (T4) Relations

32

