Exponents Associated with Y-Systems and their Relationship with q-Series
Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the fo...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210582 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860006081561886720 |
|---|---|
| author | Mizuno, Yuma |
| author_facet | Mizuno, Yuma |
| citation_txt | Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the formulation of the -system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type ᵣ, and prove this conjecture for (A₁,ℓ) and (Aᵣ,2) cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from q-series identities for this relationship.
|
| first_indexed | 2025-12-17T12:04:16Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 028, 42 pages
Exponents Associated with Y -Systems
and their Relationship with q-Series
Yuma MIZUNO
Department of Mathematical and Computing Science, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
E-mail: mizuno.y.aj@m.titech.ac.jp
Received September 27, 2019, in final form April 02, 2020; Published online April 18, 2020
https://doi.org/10.3842/SIGMA.2020.028
Abstract. Let Xr be a finite type Dynkin diagram, and ` be a positive integer greater than
or equal to two. The Y -system of type Xr with level ` is a system of algebraic relations,
whose solutions have been proved to have periodicity. For any pair (Xr, `), we define an
integer sequence called exponents using formulation of the Y -system by cluster algebras.
We give a conjectural formula expressing the exponents by the root system of type Xr, and
prove this conjecture for (A1, `) and (Ar, 2) cases. We point out that a specialization of this
conjecture gives a relationship between the exponents and the asymptotic dimension of an
integrable highest weight module of an affine Lie algebra. We also give a point of view from
q-series identities for this relationship.
Key words: cluster algebras; Y -systems; root systems; q-series
2020 Mathematics Subject Classification: 13F60; 17B22; 81R10
Contents
1 Introduction 2
2 Quiver mutations and Y -seed mutations 5
2.1 Quiver mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Y -seed mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Exponents determined from a pair (Xr, `) 7
3.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Quiver Q(Xr, `) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Mutation loop on Q(Xr, `) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Exponents of Q(Xr, `) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.6 Conjecture on exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Proofs of Theorem 3.9 17
4.1 (A1, `) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 (Ar, 2) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
This paper is a contribution to the Special Issue on Cluster Algebras. The full collection is available at
https://www.emis.de/journals/SIGMA/cluster-algebras.html
mailto:mizuno.y.aj@m.titech.ac.jp
https://doi.org/10.3842/SIGMA.2020.028
https://www.emis.de/journals/SIGMA/cluster-algebras.html
2 Y. Mizuno
5 Partition q-series 24
5.1 Mutation networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Definition of partition q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3 Asymptotics of partition q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4 Partition q-series of the mutation loop on Q(Xr, `) . . . . . . . . . . . . . . . . . 30
6 Relationships with affine Lie algebras 38
6.1 Affine Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Exponents and asymptotic dimension . . . . . . . . . . . . . . . . . . . . . . . . . 40
References 40
1 Introduction
In the study of thermodynamic Bethe ansatz, Zamolodchikov introduced a system of algebraic
relations called Y -system for any finite type simply laced Dynkin diagram and conjectured that
solutions of the Y -system have periodicity. This periodicity conjecture was proved by Fomin
and Zelevinsky [5]. Their proof was implicitly based on the theory of cluster algebras that they
introduced in [4]. In fact, in [6] they refined the result on the periodicity using the concept in
cluster algebras called Y -seed mutations.
More generally, for any finite type Dynkin diagram Xr (= Ar, Br, Cr, Dr, E6,7,8, F4 or G2)
and integer ` ≥ 2 called level, the Y -system associated with the pair (Xr, `) is defined in [21].
When X is ADE and the level is 2, this is the Y -system introduced by Zamolodchikov. The
periodicities for these Y -systems were also proved in [12, 13, 18]. Their proof used the quiver
representation theory, in particular the algebraic objects called the cluster categories.
Although these periodicities are themselves very interesting, it is also interesting that they
relate to conformal field theories through the Rogers dilogarithm function. The sum of special
values of the Rogers dilogarithm function associated with the Y -system gives the central charge
of a conformal field theory [12, 13, 29]. This is called the dilogarithm identities in conformal field
theories. Note that the central charge also appears as the exponential growth of a character
in the representation theory of affine Lie algebras [15]. For the proof of these identities, the
periodicity of the Y -system mentioned earlier plays an essential role.
The purpose of this paper is to introduce a sequence of integers associated with the Y -system
that we call exponents and to reveal interesting features of it. In particular, we give a conjectural
formula on the exponents that gives a new connection between the theory of cluster algebras
and the representational theory of affine Lie algebras, and give proofs for some cases.
In order to define the exponents, we first review roughly formulation of the Y -systems by
cluster algebras. First, we define a quiver Q(Xr, `) for a pair (Xr, `). Then, we can obtain
the Y -system as algebraic relations between rational functions obtained by repeating mutations,
which are certain operations defined on quivers. In particular, we can paraphrase the periodicity
of the Y -system associated with a pair (Xr, `) to the periodicity of the cluster transformation µ,
which is a rational function obtained from a mutation sequence on Q(Xr, `). Let I be the set of
vertices in Q(Xr, `). Then the cluster transformation µ is actually an I-tuple of single rational
functions in the variables y := (yi | i ∈ I), that is, µ ∈ Q(yi | i ∈ I)I. Keller [18] (for X = ADE)
and Inoue et al. [12, 13] (for X = BCFG) proved the periodicity of the Y -system by showing
that µ has the following periodicity:
µ ◦ µ ◦ · · · ◦ µ︸ ︷︷ ︸
t(`+h∨)
(y) = y,
Exponents Associated with Y -Systems and their Relationship with q-Series 3
where t is defined by
t =
1 if Xr = Ar, Dr, E6, E7 or E8,
2 if Xr = Br, Cr or F4,
3 if Xr = G2,
and h∨ is the dual Coxeter number of Xr.
Using this result, we define the exponents as follows. First, we see that there is a unique
η ∈ (R>0)I that satisfies µ(η) = η. Then, let J(y) be the Jacobian matrix of the rational
function µ:
J(y) =
(
∂µi
∂yj
(y)
)
i,j∈I
.
The periodicity of µ implies that the t
(
`+h∨
)
-th power of J(η) is the identity matrix. Thus all
the eigenvalues of J(η) are t
(
`+ h∨
)
-th root of unities. These eigenvalues can be written as
e
2πim1
t(`+h∨) , e
2πim2
t(`+h∨) , . . . , e
2πim|I|
t(`+h∨) ,
using a sequence of integers 0 ≤ m1 ≤ m2 ≤ · · · ≤ m|I| < t
(
`+ h∨
)
. We say that this sequence
m1, . . . ,m|I| is the exponents of Q(Xr, `).
We give a conjectural formula on the exponents in terms of root systems. Let ∆ be the root
system of type Xr with an inner product (· | ·) normalized as (α |α) = 2 for long roots α. Let
α1, . . . , αr be simple roots of ∆. For any a = 1, . . . , r, we define an integer ta by ta = 2/(αa |αa).
Let ∆long and ∆long be the set of the long roots and the short roots, respectively. Let ρ be the
half of the sum of the positive roots. Using these materials, we define two polynomials NXr,`(x)
and DXr,`(x) by
NXr,`(x) =
r∏
a=1
xt(`+h
∨) − 1
xt/ta − 1
, DXr,`(x) = Dlong
Xr,`
(x)Dshort
Xr,` (x),
where the polynomials Dlong
Xr,`
(x) and Dshort
Xr,`
(x) are defined by
Dlong
Xr,`
(x) =
∏
α∈∆long
(
xt − e
2πi(ρ |α)
`+h∨
)
, Dshort
Xr,` (x) =
∏
α∈∆short
(
x− e
2πi(ρ |α)
`+h∨
)
.
The following formula is the main conjecture of this paper. It says that the exponents are
obtained from the ratio of NXr,`(x) and DXr,`(x).
Conjecture 1.1. The following identity holds for the characteristic polynomial of J(η):
det(xI − J(η)) =
NXr,`(x)
DXr,`(x)
.
We prove the conjecture in the following cases. In these cases, we give explicit expressions
for eigenvectors of J(η) using known explicit expressions for η.
Theorem 1.2. Conjecture 1.1 is true in the following cases:
1) (A1, `) for all ` ≥ 2,
2) (Ar, 2) for all r ≥ 1.
4 Y. Mizuno
This conjecture gives a relationship between cluster algebras and root systems. Even more
interestingly, it relates to the representation theory of affine Lie algebras. Let us explain this.
Let g be the finite dimensional simple Lie algebra of type Xr, and ĝ be the affine Lie algebra
associated with g. It is a central extension of the loop algebra of g together with a derivation
operator d. Let L(Λ) be the integrable highest weight ĝ-module with the highest weight Λ.
The (specialized) character χΛ(q) is a q-series such that its coefficients are the multiplicities of
the eigenvalues of the derivation operator d in L(Λ). If we set q = e2πiτ , the character χΛ(q)
converges to a holomorphic function on the upper half-plane H = {τ ∈ C | Im τ > 0}. Let τ ↓ 0
denote the limit in the positive imaginary axis. Kac and Peterson [15] found that χΛ(q) has the
following asymptotics:
lim
τ↓0
χΛ
(
e2πiτ
)
e−
πic(`)
12τ = a(Λ),
for some rational number c and real number a(Λ). The real number a(Λ) is called the asymptotic
dimension of L(Λ). They also found the explicit formula of asymptotic dimensions. Using their
formula, we find that the right-hand side of our conjecture at x = 1 and the asymptotic dimension
of L(`Λ0) where Λ0 is the 0-th fundamental weight is related as follows:
NXr,`(1)
DXr,`(1)
= |P/Q|−1a(`Λ0)−2,
where P and Q are the weight lattice and the root lattice for g, respectively. Therefore, if the
conjecture is true, we can describe the asymptotic dimension using the exponents.
Another interesting feature about the exponents that we deal with in this paper is that the
relationship between the exponents and partition q-series. Partition q-series, which is defined by
Kato and Terashima [17], is a certain q-series associated with a mutation sequence. Typically,
it has the following form:
Z(q) =
∑
u∈(Z≥0)T
q
1
2
uTKu
(q)u1 · · · (q)uT
,
where T is a positive integer, K is a T × T positive definite symmetric matrix with rational
coefficients, and (q)n is defined by
(q)n =
n∏
k=1
(
1− qk
)
.
This type of q-series has been studied in various contexts, such as characters of conformal field
theories, Rogers-Ramanujan type identities, algebraic K-theory, and 3-dimensional topology [1,
2, 3, 8, 9, 11, 22, 24, 28, 32, 33, 37, 38, 39].
As explained earlier, we can obtain a mutation sequence for any pair (Xr, `) that describes
the corresponding Y -system. We can define the partition q-series Z(q) associated with this
mutation sequence. We show that the exponents and the asymptotics of this partition q-series
are related as follows:
lim
ε→0
Z
(
e−ε
)
e−
a
ε =
√
detA+
det(I − J(η))
,
where a is a positive real number and A+ is a matrix determined by the mutation sequence
that we call (the plus one of) the Neumann–Zagier matrix. This result explains the conjec-
tural relationship between the exponents and the asymptotic dimension mentioned earlier from
Exponents Associated with Y -Systems and their Relationship with q-Series 5
a viewpoint of q-series. As observed in [17], the partition q-series conjecturally coincides with
a finite sum of the string functions associated with the integrable highest ĝ-module L(`Λ0) via
the conjecture in [22]. Since Kac and Peterson [15] showed that the asymptotic dimension ap-
pears in the asymptotics of the string functions, we can obtain the relationship between the
exponents and the asymptotic dimension from the relationship between the partition q-series
and the string functions. We see that the resulting relationship is exactly our conjecture at
x = 1. This gives us a consistency between our conjecture on exponents and the known conjec-
ture on q-series, and also gives an interesting connection between the theory of cluster algebras
and the representation theory of affine Lie algebras.
Remark 1.3. The tuple η of positive real numbers plays an important role in our definition of
the exponents. We remark that there is a conjectural explicit formula of η for general (Xr, `)
described by the q-dimension of Kirillov-Reshetikhin modules [20, Conjecture 2] (see also [22,
Conjecture 14.2 and Remark 14.3]). A version of this conjecture is also found in [19]. It is not
difficult to verify this conjecture for type A. It was also proved for type D by Lee [25], for
type E6 by Gleitz [10], and for all classical types by Lee [26].
This paper is organized as follows. In Section 2, we review quiver mutations and Y -seed mu-
tations. In Section 3, we define the exponents associated with the Y -system for any pair (Xr, `),
and give a conjecture that describe the exponents in terms of the root system of type Xr. In
Section 4, we give proofs of this conjecture for (Aq, `) and (Ar, 2). In Section 5, we study par-
tition q-series. We calculate the asymptotics of partition q-series in Section 5.3, and give an
explicit formula for the partition q-series associated with any pair (Xr, `) in Section 5.4. In
Section 6, we discuss relationships between the topics discussed so far and the representation
theory of affine Lie algebras. We see that the asymptotics of the conjectural identity between
the partition q-series and the string functions yields our conjecture at x = 1. We point out that
this gives the relationship between the exponents and the asymptotic dimensions of integrable
highest weight representations of affine Lie algebras.
2 Quiver mutations and Y -seed mutations
Here, we review quiver mutations and Y -seed mutations following [6].
2.1 Quiver mutations
Set n to be a fixed positive integer. A quiver is a directed graph with vertices I := {1, . . . , n}
that may have multiple edges. In this paper, we assume that quivers do not have 1-loops and
2-cycles:
.
For example,
21
3
4
is a quiver.
6 Y. Mizuno
Definition 2.1. Let Q be a quiver, and let k be a vertex of Q. The quiver mutation µk is
a transformation that transforms Q into the quiver µk(Q) defined by the following three steps:
1. For each length two path i→ k → j, add a new arrow i→ j.
2. Reverse all arrows incident to the vertex k.
3. Remove all 2-cycles.
The transition
k −→
µk
k
is an example of a quiver mutation, where we have omitted all labels other than the vertex k.
It is sometimes convenient to identify a quiver with a skew-symmetric matrix. Let Q be
a quiver. For vertices i, j, we denote the number of arrows from i to j by Qij . Let B be the
n× n matrix whose (i, j)-entry is defined by
Bij = Qij −Qji.
Then the matrix B is skew-symmetric. Conversely, given any skew-symmetric matrix B, we can
construct a quiver Q by
Qij = [Bij ]+,
where [x]+ = max(0, x), and this gives a bijection between the set of n × n skew-symmetric
integer matrices and the set of quivers with vertices labeled with 1, . . . , n.
In the language of skew-symmetric matrices, the quiver mutationQ 7→ µk(Q) can be described
as follows:
B̃ij =
−Bij if i = k or j = k,
Bij +BikBkj if Bik > 0 and Bkj > 0,
Bij −BikBkj if Bik < 0 and Bkj < 0,
Bij otherwise,
where B and B̃ are the skew-symmetric matrices corresponding to Q and µk(Q), respectively.
Let ν be a permutation of {1, . . . , n}. We define the action of ν on a quiver Q by σ(Q)ij =
Qν−1(i)ν−1(j).
Let m = (m1, . . . ,mT ) be a sequence of vertices of a quiver Q, and ν be a permutation on
the vertices of Q. Consider the transitions of quivers
Q(0)
µm1−−→ Q(1)
µm2−−→ · · ·
µmT−−−→ Q(T )
ν−→ ν(Q(T )).
where Q(0) := Q. We say that a triple γ = (Q,m, ν) is a mutation loop if Q(0) = ν(Q(T ))
We define the following notion introduced in [30].
Definition 2.2. We say that a mutation loop γ = (Q,m, ν) is regular if it satisfies the following
conditions:
1. The set {νn(mt) |n ∈ Z, 1 ≤ t ≤ T} coincides with I.
2. The vertices m1, . . . ,mT belong to distinct ν-orbits in I.
Exponents Associated with Y -Systems and their Relationship with q-Series 7
2.2 Y -seed mutations
Let F be the field of rational functions in the variables y1, . . . , yn over Q, and let
Fsf =
{
f(y1, . . . , yn)
g(y1, . . . , yn)
∈ F
∣∣∣∣ f and g are non-zero polynomials in Q[y1, . . . , yn]
with non-negative coefficients
}
be the set of subtraction-free rational expressions in y1, . . . , yn over Q. This is closed under the
usual multiplication and addition, and is called universal semifield in the variables y1, . . . , yn.
A Y -seed is a pair (Q,Y ) where Q is a quiver with vertices {1, . . . , n}, and Y = (Y1 . . . , Yn)
is an n-tuple of elements of Fsf . Given this, Y -seed mutations are defined as follows.
Definition 2.3. Let (Q,Y ) be a Y -seed, and let k ∈ {1, . . . , n}. The Y -seed mutation µk is
a transformation that transforms (Q,Y ) into the Y -seed µk(Q,Y ) =
(
Q̃, Ỹ
)
, where Q̃ = µk(Q)
defined in Definition 2.1, and Ỹ is defined as follows:
Ỹi =
Y −1
k if i = k,
Yi
(
Y −1
k + 1
)−Qki if i 6= k, Qki ≥ 0,
Yi(Yk + 1)Qik if i 6= k, Qik ≥ 0.
Let ν be a permutation of {1, . . . , n}. We define an action of ν on a Y -seed by ν(Q,Y ) =
(ν(Q), ν(Y )) where ν(Y )i = Yν−1(i).
Let γ = (Q,m, ν) be a mutation loop, and let (Q(0), Y (0)) be the Y -seed defined by Q(0) = Q
and Y (0) = (y1, . . . , yn). Then the mutation loop γ gives the following transitions of Y -seeds:
(Q(0), Y (0))
µm1−−→ · · ·
µmT−−−→ (Q(T ), Y (T ))
ν−→ (ν(Q(T )), ν(Y (T ))). (2.1)
Although Q(0) = ν(Q(T )) holds from the definition of a mutation loop, we have Y (0) 6= ν(Y (T ))
in general. We denote ν(Y (T )) by µγ(y), and call it the cluster transformation of γ. It is an
element of (Fsf)
n.
3 Exponents determined from a pair (Xr, `)
3.1 Root systems
In this section, we introduce notations of root systems. Let ∆ be a root system of type Xr on
a R-vector space with an inner product normalized as (α |α) = 2 for long roots α, where Xr is
a finite type Dynkin diagram in the Fig. 1. Let α1, . . . , αr be simple roots, where the numberings
are consistent with the numberings of nodes in Fig. 1. Let ∆long and ∆short be the set of long
roots and short roots, respectively. Let ∆+ be the set of positive roots, and ρ be the half of the
sum of the positive roots:
ρ =
1
2
∑
α∈∆+
α.
We define an integer t by
t =
1 if Xr = Ar, Dr, E6, E7 or E8,
2 if Xr = Br, Cr or F4,
3 if Xr = G2.
(3.1)
For a = 1, . . . , r, we also define an integer ta ∈ {1, 2, 3} by
ta =
2
(αa |αa)
.
8 Y. Mizuno
Ar 1 2
· · ·
r − 1 r
Br 1 2
· · ·
r − 1 r
Cr 1 2
· · ·
r − 1 r
Dr 1 2
· · ·
r − 2 r − 1
r
E6
1 2 3 5 6
4
E7
1 2 3 4 5 6
7
E8
1 2 3 4 5 6 7
8
F4
1 2 3 4
G2
1 2
Figure 1. The list of finite type Dynkin diagrams.
3.2 Quiver Q(Xr, `)
For any finite type Dynkin diagram Xr and positive integer ` such that ` ≥ 2 (called a level), In-
oue, Iyama, Keller, Kuniba and Nakanishi [12, 13] defined a quiver Q = Q(Xr, `) and a mutation
loop γ = γ(Xr, `) on Q(Xr, `) that have the following periodicity:
µγ ◦ µγ ◦ · · · ◦ µγ︸ ︷︷ ︸
t(`+h∨)
(y) = y,
where h∨ is the dual Coxeter number of Xr. The list of dual Coxeter numbers is given by
Xr Ar Br Cr Dr E6 E7 E8 F4 G2
h∨ r + 1 2r − 1 r + 1 2r − 2 12 18 30 9 4
. (3.2)
We review the definition of Q(Xr, `) in this section, and the definition of γ(Xr, `) in the next
section. In the definition of Q(Xr, `), we also give auxiliary labels on vertices as
+ or − if X = A,D or E,
Exponents Associated with Y -Systems and their Relationship with q-Series 9
+ or −, and or if X = B,C or F ,
+,−, I, II, III, IV,V or VI, and or if X = G.
These labels will be used when defining a mutation sequence γ(Xr, `).
Type ADE
First, we assume that Xr is simply laced, that is, X = A,D or E. Let X ′r′ be another simply
laced Dynkin diagram. Let C = (Cij)i,j=1,...,r and C ′ = (Ci′j′)i′,j′=1,...,r′ be Cartan matrices of
types Xr and X ′r′ , respectively. Let I = {1, . . . , r} and I ′ = {1, . . . , r′} be the nodes in these
Dynkin diagrams. We fix bipartite decompositions I = I+t I− and I ′ = I ′+t I ′−. Let I = I× I ′.
For an element i = (i, i′) ∈ I, we write i : (++) if i ∈ I+× I ′+, etc. Let B be the skew-symmetric
I× I matrix defined by
Bij =
−Cijδi′j′ if i : (−+), j : (++) or i : (+−), j : (−−),
Cijδi′j′ if i : (++), j : (−+) or i : (−−), j : (+−),
−δijCi′j′ if i : (++), j : (+−) or i : (−−), j : (−+),
δijCi′j′ if i : (+−), j : (++) or i : (−+), j : (−−),
0 otherwise.
We define a quiver Q(Xr, `) as the quiver corresponding to the skew-symmetric matrix B for
(Xr, X
′
r′) = (Xr, A`−1). For each vertex i, we label i as + if i : (++) or (−−) and − if i : (+−)
or (−+).
For example, the quiver Q(A4, 4) is given by
`− 1
r
−
+
−
+
−
+
−
+
−
+
−
+
,
and the quiver Q(D5, 4) is given by
`− 1
−
+
−
+
−
+
−
+
−
+
−
+
+
−
+
.
Type BCF
Next, we assume that X = B,C or F .
10 Y. Mizuno
We define Q(Br, `) as
`− 1
r − 1 r − 1
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
+
−
+
−
+
−
+
−
+
if ` is even, and
`− 1
r − 1 r − 1
· · ·
· · ·
· · ·
· · ·
−
+
+
−
−
+
+
−
+
−
+
+
−
−
+
+
−
if ` is odd.
We define Q(Cr, `) as
2`− 1
r − 1
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
`− 1
+
−
+
−
+
−
+
−
+
−
+
−
+
if ` is even, and
2`− 1
r − 1
· · ·
· · ·
· · ·
· · ·
· · ·
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
`− 1
+
−
+
−
+
−
+
+
−
Exponents Associated with Y -Systems and their Relationship with q-Series 11
if ` is odd, where we identify the rightmost columns in the left quivers with the center columns
in the right quivers.
We define Q(F4, `) as
2`− 1
−
+
−
+
−
+
−
+
−
+
−
+
−
+
`− 1
+
−
+
−
+
−
+
−
+
−
+
−
+
+
−
+
−
+
−
if ` is even, and
2`− 1
−
+
−
+
−
+
−
+
−
+
`− 1
+
−
+
−
+
−
+
+
−
+
−
−
+
if ` is odd, where we identify the rightmost columns in the left quivers with the center columns
in the right quivers.
Type G
Finally, we assume that Xr = G2. We define Q(G2, `) as
`− 1
IV
I
IV
+
−
+
−
+
−
+
−
+
−
+
II
V
II
+
−
+
−
+
−
+
−
+
−
+
VI
III
VI
+
−
+
−
+
−
+
−
+
−
+
3`− 1
12 Y. Mizuno
if ` is even, and
`− 1
I
IV
+
−
+
−
+
−
+
−
V
II
+
−
+
−
+
−
+
−
III
VI
+
−
+
−
+
−
+
−
3`− 1
if ` is odd, where we identify the three columns consisting of filled vertices.
3.3 Mutation loop on Q(Xr, `)
In this section, we define a mutation loop γ(Xr, `) on the quiver Q(Xr, `).
Type ADE
First, we assume that Xr is simply laced, that is, X = A,D or E. Let I be the set of the vertices
in Q(Xr, `), and I+ and I− be the subsets of I consisting of the vertices labeled with + and −,
respectively. We define the following two compositions of mutations:
µ+ =
∏
m∈I+
µm, µ− =
∏
m∈I−
µm.
We can easily verify the following lemma.
Lemma 3.1. Let Q = Q(Xr, `). Consider the following transitions of quivers:
Q
µ+−−→ Q′
µ−−−→ Q′′.
Then Q′ and Q′′ do not depend on orders of the mutations in µ+ and µ−. Furthermore, we have
Q = Q′′.
Thus
γ(Xr, `) := ((I+, I−), id)
is a mutation loop, where I+ and I− in this equation mean any sequence in which all the elements
in I+ and I−, respectively, appear exactly once.
Type BCF
Next, we assume that X = B,C or F . Let I be the set of the vertices in Q(Xr, `), and I+,
I−, I+ and I− be the subsets of I consisting of the vertices labeled with (+, ), (−, ), (+, ) and
(−, ), respectively. We define the following two compositions of mutations:
µ+µ+ =
∏
m∈I+tI+
µm, µ− =
∏
m∈I−
µm.
Let ν be the permutation of vertices in Q(Xr, `) that is identity on the filled vertices, and
left-right reflection on the unfilled vertices.
Exponents Associated with Y -Systems and their Relationship with q-Series 13
Lemma 3.2 ([12, 13]). Let Q = Q(Xr, `). Consider the following transitions of quivers:
Q
µ+µ+−−−→ Q′
µ−−−→ Q′′.
Then Q′ and Q′′ do not depend on orders of the mutations in µ+µ+ and µ−. Furthermore, we
have Q = ν(Q′′).
Thus
γ(Xr, `) :=
((
I+ t I+, I−
)
, ν
)
is a mutation loop, where I+ t I+ and I− in this equation mean any sequence in which all the
elements in I+ t I+ and I−, respectively, appear exactly once.
Type G
Next, we assume that X = G. Let I be the set of the vertices in Q(G2, `). Let I+, I− be the
subsets of I consisting of the vertices labeled with (+, ) and (−, ) respectively, and let II , . . . , IVI
be the subsets of I consisting of the vertices labeled with (I, ), . . . , (VI, ), respectively. We define
the following two compositions of mutations:
µ+µI =
∏
m∈I+tII
µm, µ−µII =
∏
m∈I−tIII
µm.
We choose indices of vertices in Q(G2, `) to be
{(a, i) | a = 1, 2, 3, i = 1, . . . , `− 1} ∪ {(4, i) | i = 1, . . . , 3`− 1},
so that (a, i) is the i-th row (from the bottom) vertex in the a-th quiver (from the left). Let ν
be the permutation of the vertices in Q(G2, `) that is the identity on the filled vertices, that is,
ν(4, i) = (4, i) for all i, and acts on the unfilled vertices as
ν(1, i) = (2, i),
ν(2, i) = (3, i),
ν(3, i) = (1, i),
for all i.
Lemma 3.3 ([13]). Let Q = Q(G2, `). Consider the following transitions of quivers:
Q
µ+µI−−−→ Q′
µ−µII−−−→ Q′′.
Then Q′ and Q′′ do not depend on orders of the mutations in µ+µI and µ−µII. Furthermore,
we have Q = ν(Q′′).
Thus
γ(G2, `) :=
((
I+ t II , I− t III
)
, ν
)
is a mutation loop, where I+ t II and I− t III in this equation mean any sequence in which all
the elements in I+ t II and I− t III, respectively, appear exactly once.
14 Y. Mizuno
3.4 Periodicity
As above, we have defined quiver Q(Xr, `) and the mutation loop γ(Xr, `) for any pair (Xr, `).
The following lemma follows from these definitions.
Lemma 3.4. The mutation loop γ(Xr, `) is regular.
A remarkable fact, as stated in the beginning of Section 3.2, is that these mutation loops
have periodicity.
Theorem 3.5 ([12, 13, 18]). Let γ = γ(Xr, `). Then the following holds:
µγ ◦ µγ ◦ · · · ◦ µγ︸ ︷︷ ︸
t(`+h∨)
(y) = y,
where h∨ is the dual Coxeter number of Xr (see (3.2) for the list of dual Coxeter numbers),
and t is given by (3.1).
3.5 Exponents of Q(Xr, `)
First, we see that the rational function µγ for the mutation loop γ = γ(Xr, `) satisfies the
following property.
Proposition 3.6. The fixed point equation µγ(y) = y has a unique positive real solution.
This lemma will be proved in Section 5.3 (see Corollary 5.16). This positive real solution is
denoted by η ∈ (R>0)I.
Let J(y) be the Jacobian matrix of the rational function µγ :
Jγ(y) :=
(
∂µi
∂yj
(y)
)
i,j∈I
,
where µi is the i-th components of µγ . Then Theorem 3.5 implies that the t
(
` + h∨
)
-th power
of Jγ(η) is the identity matrix:
Jγ(η)t(`+h
∨) = I.
Thus all the eigenvalues of Jγ(η) are t
(
` + h∨
)
-th root of unities. These eigenvalues can be
written as
e
2πim1
t(`+h∨) , e
2πim2
t(`+h∨) , . . . , e
2πim|I|
t(`+h∨) ,
using a sequence of integers 0 ≤ m1 ≤ m2 ≤ · · · ≤ m|I| < t
(
`+ h∨
)
. We say that this sequence
m1, . . . ,m|I| is the exponents of Q(Xr, `).
Example 3.7. Let (Xr, `) = (A1, 4). In this case, the index set of the quiver is given by I =
I × I ′ = {(1, 1), (1, 2), (1, 3)}. We choose bipartite decompositions of I = {1} and I ′ = {1, 2, 3}
as I = I+ t I− = { } t {1} and I ′ = I ′+ t I ′− = {1, 3} t {2}, respectively. Then we obtain the
decomposition I = I+ t I− = {(1, 2)} t {(1, 1), (1, 3)}, and the mutation loop γ(A1, 4) is given
in Fig. 2, where we simply write (1, i′) as i′ in the figure.
The cluster transformation is now given by
µ1 = y−1
1 y−1
2 (y2 + 1), µ2 = y1y2y3(y1y2 + y2 + 1)−1(y2y3 + y2 + 1)−1,
µ3 = y−1
2 y−1
3 (y2 + 1),
Exponents Associated with Y -Systems and their Relationship with q-Series 15
µ+
−→
1
−
2
+
3
−
µ−
−→
1
−
2
+
3
−
1
−
2
+
3
−
Figure 2. The mutation loop γ(A1, 4).
where µi is the i-th component of µγ(y1, y2, y3). We can directly verify that η = (2, 1/3, 2) is
the positive real solution of µi = yi (i = 1, 2, 3).
Although the matrix Jγ(y) is a little complicated to write down directly, we can see that
the matrix L(y) defined by L(y) = diag(µ1, µ2, µ3)−1Jγ(y) diag(y1, y2, y3) can be written as
a product of two simple matrices as follows:
L(y) =
−1 0 0
y2 + 1
y1y2 + y2 + 1
1
y2 + 1
y2y3 + y2 + 1
0 0 −1
1
1
y2 + 1
0
0 −1 0
0
1
y2 + 1
1
.
By substituting η, we obtain
L(η) =
−1 0 0
2
3 1 2
3
0 0 −1
1 3
4 0
0 −1 0
0 3
4 1
=
−1 −3
4 0
2
3 0 2
3
0 −3
4 −1
,
and
det(xI3 − Jγ(η)) = det(xI3 − L(η)) = x3 + 2x2 + 2x+ 1
=
(
x− e
2πi·2
6
)(
x− e
2πi·3
6
)(
x− e
2πi·4
6
)
.
Thus the exponents of Q(A1, 4) are 2, 3, 4.
3.6 Conjecture on exponents
We give a conjectural formula on exponents in terms of root systems. We define two polyno-
mials NXr,`(x) and DXr,`(x) by
NXr,`(x) =
r∏
a=1
xt(`+h
∨) − 1
xt/ta − 1
, DXr,`(x) = Dlong
Xr,`
(x)Dshort
Xr,` (x),
where the polynomials Dlong
Xr,`
(x) and Dshort
Xr,`
(x) are defined by
Dlong
Xr,`
(x) =
∏
α∈∆long
(
xt − e
2πi(ρ |α)
`+h∨
)
, Dshort
Xr,` (x) =
∏
α∈∆short
(
x− e
2πi(ρ |α)
`+h∨
)
.
When X = A,D or E, the polynomials NXr,`(x) and DXr,`(x) can be written more simply:
NXr,`(x) =
(
x`+h
∨ − 1
x− 1
)r
, DXr,`(x) =
∏
α∈∆
(
x− e
2πi(ρ |α)
`+h∨
)
.
16 Y. Mizuno
1 2 3 4 5 6
Figure 3. The underlying white circles represent the exponents of NA3,3(x), and the black marks repre-
sent the exponents of DA3,3(x). The number of vertices in a same vertical line is a multiplicity. Further-
more, the black circles and diamonds represent the exponents that come from the positive roots and the
negative roots, respectively. The unmarked white circles represent the exponents of NA3,3(x)/DA3,3(x).
Conjecture 3.8. Let Xr be a finite type Dynkin diagram, and ` be a positive integer such that
` ≥ 2. Let γ = γ(Xr, `) be the mutation loop on the quiver Q(Xr, `) defined in Section 3.3. Then
the following identity holds for the characteristic polynomial of Jγ(η):
det(xI − Jγ(η)) =
NXr,`(x)
DXr,`(x)
. (3.3)
Theorem 3.9. Conjecture 3.8 is true in the following cases:
1) (A1, `) for all ` ≥ 2,
2) (Ar, 2) for all r ≥ 1.
We will prove Theorem 3.9 in Section 4.1.
3.7 Examples
In this section, we give examples of calculations on the right-hand side in the conjectural for-
mula (3.3).
Let F (x) be a polynomial whose roots are all N -th roots of unity. The roots of F (x) can be
written as
e2πim1/N , e2πim2/N , . . . , e2πimn/N ,
where 0 ≤ m1 ≤ m2 ≤ · · · ≤ mn < N is a sequence of integers. We call this sequence of integers
the exponents of F (x). The exponents of F (x) is denoted by E(F (x)).
Example 3.10. Let (Xr, `) = (A3, 3). The dual Coxeter number is given by h∨ = 4, so
`+ h∨ = 7. Therefore, the exponents of NA3,3(x) is given by
E(NA3,3(x)) = (1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6).
On the other hand, by calculation on the root system of type A3, we obtain
E(DA3,3(x)) = (1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 6).
As the result, the exponents of NA3,3(x)/DA3,3(x) are given by
E
(
NA3,3(x)
DA3,3(x)
)
= (2, 3, 3, 4, 4, 5).
Fig. 3 illustrates these exponents.
Exponents Associated with Y -Systems and their Relationship with q-Series 17
1 2 3 4 5 6 7 8 9 10 11 12 13
Figure 4. The exponents of NB3,2(x) and DB3,2(x). The meanings of the symbols are the same as in
Fig. 3.
Example 3.11. Let (Xr, `) = (B3, 2). The dual Coxeter number is given by h∨ = 5, so
t
(
`+ h∨
)
= 14. Therefore, the exponents of NB3,2(x) is given by
E(NB3,2(x)) = (1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9,
10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13).
On the other hand, by calculation on the root system of type B3, we obtain
E
(
Dlong
B3,2
(x)
)
= (1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13),
E
(
Dshort
B3,2 (x)
)
= (1, 3, 5, 9, 11, 13).
As the result, the exponents of NB3,2(x)/DB3,2(x) are given by
E
(
NB3,2(x)
DB3,2(x)
)
= (2, 4, 6, 7, 8, 10, 12).
Fig. 4 illustrates these exponents.
4 Proofs of Theorem 3.9
4.1 (A1, `) case
Let ` be an integer with ` ≥ 2. In this section we consider the mutation loop γ = γ(A1, `).
Let I = {1, . . . , ` − 1} denotes the vertices in Q(A1, `). Let us fix a bipartite decomposition
I = I+ t I−. For example, Fig. 2 shows the mutation loop γ(A1, 4).
Lemma 4.1. The right-hand side of (3.3) is given by
NA1,`(x)
DA1,`(x)
=
∏̀
a=2
(
x− e
2πia
`+2
)
.
Proof. First note that the dual Coxeter number of A1 is given by h∨ = 2. Then, the lemma
follows from
NA1,`(x) =
x`+2 − 1
x− 1
,
and
DA1,`(x) =
(
x− e
2πi
`+2
)(
x− e
−2πi
`+2
)
=
(
x− e
2πi
`+2
)(
x− e
2πi(`+1)
`+2
)
. �
18 Y. Mizuno
Therefore, what we have to show is that the exponents of Q(A1, `) are 2, 3, . . . , `. To prove
this, we define the matrix L(y) by
L(y) =
µ1(y)
. . .
µ`−1(y)
−1
Jγ(y)
y1
. . .
y`−1
,
where µi is the i-th component of µγ . The matrices Jγ(η) and L(η) have the same characteristic
polynomial because µ(η) = η. We will find the eigenvalues of L(η) instead of Jγ(η).
Let ζ = eπi/(`+2). For m = 1, 2, . . . , `− 1, we define non-zero real numbers
zm =
sin πm
`+2 sin π(m+2)
`+2
sin2 π(m+1)
`+2
=
(
ζm − ζ−m
)(
ζm+2 − ζ−m−2
)(
ζm+1 − ζ−m−1
)2 . (4.1)
We define the two matrices L+ =
(
Lmk+
)
m,k=1,...,`−1
and L− =
(
Lmk−
)
m,k=1,...,`−1
by
Lmk± =
{
δmk if k ∈ I∓,
−δmk + zk(δm,k−1 + δm,k+1) if k ∈ I±,
where δ is the Kronecker delta.
Lemma 4.2. The matrix L(η) is expressed as
L(η) = L−L+.
Proof. Let
(Q, y)
µ+−−→ (Q′, y′)
µ−−−→ (Q′′, y′′)
be the Y -seed transitions associated with γ. Then the chain rule for Jacobian matrices implies
that
L(y) = L−(y′)L+(y),
where L±(Y ) for Y ∈ (Fsf)
`−1 is the (`− 1)× (`− 1) matrices defined by
Lmk± (Y ) =
δmk if k ∈ I∓,
−δmk +
1
1 + Yk
(δm,k−1 + δm,k+1) if k ∈ I±.
Let η̃ be the (`−1)-tuple of real numbers such that its m-th entry is η if m ∈ I+ and is y′|y=η
if m ∈ I−. The following explicit expression of η̃ is known (see [23, Example 5.3] for example):
η̃m =
sin2 π
`+2
sin πm
`+2 sin π(m+2)
`+2
.
We can easily check that zm = 1/(1 + η̃m), and this complete the proof. �
For any non-zero complex number λ, we consider the following difference equation of the
numbers (φm)0≤m≤`:
φm−1 + φm+1 = φmz
−1
m
(
λ+ λ−1
)
(4.2)
with the boundary conditions given by
φ0 = φ` = 0. (4.3)
Exponents Associated with Y -Systems and their Relationship with q-Series 19
Lemma 4.3. Let (φm)0≤m≤` be a non-zero solution of the difference equation (4.2) satisfying
the boundary conditions (4.3). Then the vector
ψ =
ψ1
...
ψ`−1
defined by
ψm =
{
λφm if m ∈ I+,
φm if m ∈ I−,
is an eigenvector of the transpose matrix of L(η) with an eigenvalue λ2, that is,
L(η)Tψ = λ2ψ.
Proof. Let ψ′ = LT
−ψ and ψ′′ = LT
+ψ
′ (=L(η)Tψ
)
. In the following equations, we assume that
ψ0 = ψ` = ψ′0 = ψ′` = 0. Then we obtain
ψ′m =
{
ψm if m ∈ I+,
−ψm + zm(ψm−1 + ψm+1) if m ∈ I−,
and
ψ′′m =
{
−ψ′m + zm(ψ′m−1 + ψ′m+1) if m ∈ I+,
ψ′m if m ∈ I−.
For any m ∈ I−, we compute
ψ′′m − λ2ψm = ψ′m − λ2ψm = −ψm + zm(ψm−1 + ψm+1)− λ2ψm
= −φm + zmλ(φm−1 + φm+1)− λ2φm
= zmλ
(
−z−1
m
(
λ+ λ−1
)
φm + φm−1 + φm+1
)
= 0.
In particular, we obtain ψ′m = λ2ψm for m ∈ I−. Thus, for any m ∈ I+, we find that
ψ′′m − λ2ψm = −ψ′m + zm(ψ′m−1 + ψ′m+1)− λ2ψm = −ψm + zmλ
2(ψm−1 + ψm+1)− λ2ψm
= −λφm + zmλ
2(φm−1 + φm+1)− λ3φm
= zmλ
2
(
−z−1
m
(
λ+ λ−1
)
φm + φm−1 + φm+1
)
= 0,
and this complete the proof. �
Now we focus on solving the difference equation (4.2) satisfying the boundary conditions (4.3).
We will show that
(
φ
(a)
m
)
m=0,...,`
for
φ(a)
m = det
2 cos
πa
`+ 2
2 cos
πa(m+ 1)
`+ 2
sin
π(a− 1)
`+ 2
/sin
π
`+ 2
sin
π(a− 1)(m+ 1)
`+ 2
/sin
π(m+ 1)
`+ 2
(4.4)
is a non-zero solution of (4.2) and (4.3) for λ = ζa if a = 2, . . . , ` (Theorem 4.7).
To prove this, we introduce the following Laurent polynomials:
α(a)(x) = xa + x−a, β(a)(x) =
xa−1 − x−a+1
x− x−1
.
20 Y. Mizuno
We write α(a)
(
xm+1
)
and β(a)
(
xm+1
)
as α
(a)
m (x) and β
(a)
m (x). We also define a Laurent polyno-
mial P
(a)
m (x) by
P (a)
m (x) = det
(
α
(a)
0 (x) α
(a)
m (x)
β
(a)
0 (x) β
(a)
m (x)
)
.
Note that φ
(a)
m in (4.4) can be written as P
(a)
m (ζ).
Let us examine difference equations for α
(a)
m (x), β
(a)
m (x) and P
(a)
m (x).
Lemma 4.4. The Laurent polynomials α
(a)
m (x) satisfy the difference equation
α
(a)
m−1(x) + α
(a)
m+1(x) = α
(a)
0 (x)α(a)
m (x). (4.5)
Proof. We compute
α
(a)
m−1 + α
(a)
m+1(x) =
(
xam + x−am
)
+
(
xa(m+2) + x−a(m+2)
)
=
(
xa + x−a
)(
xa(m+1) + x−a(m+1)
)
= α
(a)
0 (x)α(a)
m (x),
and this proves the lemma. �
Lemma 4.5. The Laurent polynomials β
(a)
m (x) satisfy the difference equation(
β
(a)
m−1(x) + β
(a)
m+1(x)
)
(xm − x−m)
(
xm+2 − x−m−2
)
= α
(a)
0 (x)β(a)
m (x)
(
xm+1 − x−m−1
)2 − α(a)
m (x)β
(a)
0 (x)
(
x− x−1
)2
. (4.6)
Proof. We compute(
β
(a)
m−1(x) + β
(a)
m+1(x)
)(
xm − x−m)(xm+2 − x−m−2
)
− α(a)
0 (x)β(a)
m (x)
(
xm+1 − x−m−1
)2
=
(
x(a−1)m − x−(a−1)m
)(
xm+2 − x−m−2
)
+
(
x(a−1)(m+2) − x−(a−1)(m+2)
)(
xm − x−m
)
− α(a)
0 (x)
(
x(a−1)m − x−(a−1)m
)(
xm+1 − x−m−1
)
=
(
xam+2 + x−am−2 − xam−2m−2 − x−am+2m+2
)
+
(
xa(m+2)−2 + x−a(m+2)+2 − xa(m+2)−2m−2 − x−a(m+2)+2m+2
)
−
(
xa(m+2) + xam − x−am+2m+2 − x−a(m+2)+2m+2
− xa(m+2)−2m−2 − xam−2m−2 + x−am + x−a(m+2)
)
= xam+2 + x−am−2 + xa(m+2)−2 + x−a(m+2)+2 − xam − x−am − xa(m+2) − x−a(m+2)
=
(
x− x−1
)(
xam+1 − x−am−1 + xa(m+2)−1 − x−a(m+2)+1
)
= −
(
x− x−1
)(
xa(m+1) + x−a(m+1)
)(
xa−1 − x−a+1
)
= −α(a)
m (x)β
(a)
0 (x)
(
x− x−1
)2
,
and this proves the lemma. �
Lemma 4.6. The Laurent polynomials P
(a)
m (x) satisfy the difference equation(
P
(a)
m−1(x) + P
(a)
m+1(x)
)(
xm − x−m
)(
xm+2 − x−m−2
)
= P (a)
m (x)α
(a)
0 (x)
(
xm+1 − x−m−1
)2
.
Exponents Associated with Y -Systems and their Relationship with q-Series 21
Figure 5. The values of
(
φ
(2)
m
)
m=0...,`
and
(
φ
(3)
m
)
m=0...,`
for ` = 5.
Proof. By using (4.5), (4.6) and the relation(
xm+1 − x−m+1
)2 − (x− x−1
)2
=
(
xm − x−m
)(
xm+2 − x−m−2
)
,
we obtain
P (a)
m (x)α
(a)
0 (x)
(
xm+1 − x−m−1
)2
=
(
α
(a)
0 (x)β(a)
m (x)− α(a)
m (x)β
(a)
0 (x)
)
α
(a)
0 (x)
(
xm+1 − x−m−1
)2
= α
(a)
0 (x)
((
β
(a)
m−1(x) + β
(a)
m+1(x)
)(
xm − x−m
)(
xm+2 − x−m−2
)
+ α(a)
m (x)β
(a)
0 (x)
(
x− x−1
)2)− α(a)
0 (x)α(a)
m (x)β
(a)
0 (x)
(
xm+1 − x−m−1
)2
= α
(a)
0 (x)
(
β
(a)
m−1(x) + β
(a)
m+1(x)
)(
xm − x−m
)(
xm+2 − x−m−2
)
− α(a)
0 (x)α(a)
m (x)β
(a)
0 (x)
((
xm+1 − x−m+1
)2 − (x− x−1
)2)
=
(
α
(a)
0 (x)
(
β
(a)
m−1(x) + β
(a)
m+1(x)
)
− α(a)
0 (x)α(a)
m (x)β
(a)
0 (x)
)
×
(
xm − x−m
)(
xm+2 − x−m−2
)
=
(
α
(a)
0 (x)
(
β
(a)
m−1(x) + β
(a)
m+1(x)
)
−
(
α
(a)
m−1(x) + α
(a)
m+1(x)
)
β
(a)
0 (x)
)
×
(
xm − x−m
)(
xm+2 − x−m−2
)
=
(
P
(a)
m−1(x) + P
(a)
m+1(x)
)(
xm − x−m
)(
xm+2 − x−m−2
)
,
completing the proof. �
Using Lemma 4.6, we can construct non-zero solutions of the difference equation (4.2) satis-
fying the boundary conditions (4.3).
Theorem 4.7. For any a = 2, 3, . . . , ` and m = 0, 1, . . . , `, let φ
(a)
m = P
(a)
m (ζ). Then the
following properties hold:
1. The numbers
{
φ
(a)
m | 0 ≤ m ≤ `
}
satisfy the difference equation (4.2) for λ = ζa.
2. The boundary conditions (4.3) hold, i.e., φ
(a)
0 = φ
(a)
` = 0.
3. There exists m such that φ
(a)
m 6= 0.
Proof. The property (1) follows from Lemma 4.6 and the definition of zm (4.1). Now we prove
the property (2). The definition of P
(a)
m (x) immediately implies that P
(a)
0 (x) = 0, hence φ
(a)
0 = 0.
To prove φ
(a)
` = 0, we define a Laurent polynomial P̃ (a)(x, y) of two variables by
P̃ (a)(x, y) = det
(
α(a)(x) α(a)(y)
β(a)(x) β(a)(y)
)
.
22 Y. Mizuno
Figure 6. The values of
(
φ
(4)
m
)
m=0...,`
and
(
φ
(5)
m
)
m=0...,`
for ` = 5.
It is easy to see that the polynomial P̃ (a)(x, y) satisfies
P̃ (a)(x, xm+1) = P (a)
m (x),
P̃ (a)(x, y−1) = P̃ (a)(x, y),
P̃ (a)(x,−y) = (−1)aP̃ (a)(x, y).
Thus we have
φ
(a)
` = P
(a)
` (ζ) = P̃ (a)
(
ζ, ζ`+1
)
= P̃ (a)
(
ζ,−ζ−1
)
= (−1)aP̃ (a)
(
ζ, ζ−1
)
= (−1)aP̃ (a)(ζ, ζ) = 0.
Now we prove the property (3). We will show that φ
(a)
1 > 0. The number φ
(a)
1 can be written as
φ
(a)
1 = ζa + ζ−a · ζ
2a−2 − ζ−2a+2
ζ2 − ζ−2
− ζ2a + ζ−2a · ζ
a−1 − ζ−a+1
ζ − ζ−1
=
ζa−1 − ζ−a+1
ζ2 − ζ−2
((
ζa + ζ−a
)(
ζa−1 + ζ−a+1
)
−
(
ζ2a + ζ−2a
)(
ζ + ζ−1
))
=
ζa−1 − ζ−a+1
ζ2 − ζ−2
((
ζ + ζ−1
)
−
(
ζ2a+1 + ζ−2a−1
))
= sin
(a− 1)π
`+ 2
(
sin
2π
`+ 2
)−1(
2 cos
π
`+ 2
− 2 cos
(2a+ 1)π
`+ 2
)
.
This shows that φ
(a)
1 > 0 for a = 2, 3, . . . , `. �
We plot the values of
(
φ
(a)
m
)
m=0...,`
for ` = 5 and a = 2, 3, 4, 5 in Figs. 5 and 6. The underlying
graphs are plots of the function
det
2 cos
πa
`+ 2
2 cos
πa(u+ 1)
`+ 2
sin
π(a− 1)
`+ 2
/
sin
π
`+ 2
sin
π(a− 1)(u+ 1)
`+ 2
/
sin
π(u+ 1)
`+ 2
in an interval 0 ≤ u ≤ `, and the points on these graphs represent the values of
(
φ
(a)
m
)
m=0...,`
.
Corollary 4.8. The exponents of Q(A1, `) are 2, 3, . . . , `, that is,
det(xI − Jγ(η)) =
∏̀
a=2
(
x− e
2πia
`+2
)
.
In particular, Conjecture 3.8 is true in this case.
Exponents Associated with Y -Systems and their Relationship with q-Series 23
Proof. From Theorem 4.7, we find that e
2πia
`+2 for a = 2, 3, . . . , ` are eigenvalues of Jγ(η). These
are all the eigenvalues and their multiplicities are one, since the size of Jγ(η) is `− 1. �
4.2 (Ar, 2) case
First, we will see that the right-hand side of the conjectural formula (3.3) for (Ar, 2) is the same
as that for (A1, `) if we change the parameter as r ↔ ` − 1. Such a phenomenon is known as
the level-rank duality.
Lemma 4.9. The right-hand side in (3.3) is given by
NAr,2(x)
DAr,2(x)
=
r+1∏
a=2
(
x− e
2πia
r+3
)
.
Proof. Because h∨ = r + 1, we obtain
NAr,2(x) =
(
xr+3 − 1
x− 1
)r
,
and
DAr,2(x) =
∏
α∈∆
(
x− e
2πi(ρ |α)
r+3
)
.
We can easily see that, say using a concrete realization of the root system of type Ar, the
following holds:
#{α ∈ ∆+ | (ρ |α) = a} =
{
r + 1− a if 1 ≤ a ≤ r + 1,
0 if a = r + 2.
Thus for any 1 ≤ a ≤ r + 2, we obtain
#{α ∈ ∆+ | (ρ |α) = a mod r + 3}
= #{α ∈ ∆+ | (ρ |α) = a}+ #{α ∈ ∆+ | (ρ | − α) = a− r − 3}
=
r if a = 1,
r − 1 if 2 ≤ a ≤ r + 1,
r if a = r + 2,
and this implies that
DAr,2(x)
r+1∏
a=2
(
x− e
2πia
r+3
)
= NAr,2(x),
completing the proof. Fig. 7 illustrates these calculations. �
We now complete the proof of Theorem 3.9.
Corollary 4.10. Conjecture 3.8 is true for (Ar, 2).
Proof. When we choose suitable bipartite decompositions, the mutation loop γ = γ(Ar, 2) is
obtained from the mutation loop γ′ = γ(A1, r + 1) by reversing all arrows in quivers. This
implies that their cluster transformations are related by µγ′ = ι ◦ µγ ◦ ι, where ι is the map
(y1, . . . , yr) 7→
(
y−1
1 , . . . , y−1
r
)
. From this relation, we can easily verify that the exponents
for (Ar, 2) are the same as those for (A1, r + 1). Thus the corollary follows from Corollary 4.8
and Lemma 4.9. �
24 Y. Mizuno
1 2 3 4 5 6 7 8
Figure 7. The exponents of NAr,2 and DAr,2 for r = 6. The meanings of the symbols are the same as
in Fig. 3.
5 Partition q-series
5.1 Mutation networks
For a mutation loop γ = (Q,m, ν), we define a combinatorial object called a mutation network,
which we will denote by Nγ . This was introduced in [34]. Roughly speaking, a mutation network
is a graph obtained by extracting only the parts that change in the transitions of the quivers.
We will use mutation networks to define partition q-series.
Consider the sets defined by
E(t) = {(i, t) | 1 ≤ i ≤ n}, E =
⋃
0≤t≤T
E(t).
We define an equivalence relation ∼ on E as follows:
1. For all t = 1, . . . , T , (i, t− 1) ∼ (i, t) if i 6= mt.
2. For all i = 1 . . . , n, (ν(i), T ) ∼ (i, 0).
Let E = E/ ∼ be the corresponding quotient set.
First, we define graphs N γ(t) for all t = 1, . . . , T . These consist of two types of vertices: black
vertices and a single square vertex. The black vertices are labeled with elements in E(t−1)∪E(t),
and represented as solid circles , while the square vertex is labeled with t, and represented as a
hollow square . There are three types of edges joining these vertices: broken edges, arrows from
the square vertex to a black vertex, and arrows from a black vertex to the square vertex. These
edges are added as follows. First, we add broken edges between the square vertex and black
vertices labeled with (mt, t− 1) or (mt, t). Then, for all i = 1, . . . , n, we add Qmt,i(t− 1) arrows
from the square vertex to the black vertex labeled with (i, t− 1) if Qmt,i(t− 1) > 0. Finally, for
all i = 1, . . . , n, we add Qi,mt(t − 1) arrows from the black vertex labeled with (i, t − 1) to the
square vertex if Qi,mt(t− 1) > 0. These rules can be summarized as follows:
1)
(i, s) t
if (i, s) = (mt, t− 1) or (mt, t),
2)
(i, s) a t
if a = Qmt,i(t− 1) > 0 and s = t− 1, for all i,
Exponents Associated with Y -Systems and their Relationship with q-Series 25
3)
(i, s) a t
if a = Qi,mt(t− 1) > 0 and s = t− 1, for all i.
Let N γ = ∪1≤t≤TN γ(t). Then the mutation network Nγ of the mutation sequence γ is
defined to be the quotient of N γ by ∼:
Nγ = N γ/ ∼ .
Here, the quotient carried out such that the vertices are labeled with the elements of E, and
all edges are included (i.e., we do not cancel the arrows pointing in the opposite direction). We
denote the set of square vertices of Nγ by ∆, that is, ∆ = {1, . . . , T}.
For a mutation network Nγ , we define a map ϕ : ∆→ E from the set of the black vertices to
the set of the square vertices by
ϕ(t) = [(mt, t− 1)],
where [(mt, t− 1)] is an equivalence class that contains (mt, t− 1). Let I/ν is the set of ν-orbits
in I:
I/ν =
{{
νk(i) | k ∈ Z
}
| i ∈ I
}
.
We also define a map ψ : ∆→ I/ν by
ψ(t) = [mt],
where [mt] is the ν-orbit of mt.
Lemma 5.1. If γ is regular, both the map ϕ and ψ defined above are bijections.
Proof. The injectivity of ϕ follows from the definition of E and ϕ, and the surjectivity of ϕ
follows from the condition (1) in Definition 2.2. The injectivity of ψ follows from the condition (1)
in Definition 2.2, and the surjectivity of ψ follows from the condition (2) in Definition 2.2. �
In particular, the number of black vertices and the number of square vertices for a regular
mutation loop are the same since ϕ is a bijection.
Example 5.2. Define a quiver Q by
Q = 1 2.
Let γ = (Q,m, ν), where m = (1, 2) and ν = id. It is a regular mutation loop since the quiver
transitions are given by
1 2 −→µ1 1 2 −→µ2 1 2,
where the underlines indicate the mutated vertices. The graphs N γ(t) can then be given as
N γ(1) =
(1, 0)
1
(1, 1)(2, 1)
(2, 0)
, N γ(2) =
(1, 1)
2
(2, 1)
(1, 2)(2, 2)
.
26 Y. Mizuno
Let E = {e1, e2} be the set of black vertices in Nγ , where
e1 = {(1, 0), (1, 1), (1, 2)}, e2 = {(2, 0), (2, 1), (2, 2)}.
The mutation network of γ is then given by
Nγ =
e1 e2
1 2
. (5.1)
Definition 5.3. Let γ be a mutation loop and Nγ be its mutation network. For any given pair
consisting of a black vertex labeled with e ∈ E and a square vertex labeled with t ∈ ∆, let N et
0
be the number of broken lines between e and t, N et
+ be the number of arrows from t to e, and N et
−
be the number of arrows from e to t:
N et
0 = #
{ e t }
, N et
+ = #
{ e t }
, N et
− = #
{ e t }
.
Let N0 be the E ×∆ matrix whose (e, t)-entry is N et
0 . Define the E ×∆ matrices N+ and N−
likewise. We say that the matrices N0, N+, and N− are the adjacency matrices of the mutation
network Nγ .
Definition 5.4. Let N0, N+, and N− be the adjacency matrices of the mutation network Nγ .
Define two additional E ×∆ matrices A+ =
(
Aet+
)
and A− =
(
Aet−
)
by
A+ = N0 −N+, A− = N0 −N−.
We call the matrices A+ and A− the Neumann–Zagier matrices of the mutation loop γ.
Example 5.5. Let γ be the mutation loop defined in Example 5.2. It follows from (5.1) that
the adjacency matrices of the mutation network Nγ are given by
N0 =
(
2 0
0 2
)
, N+ =
(
0 0
0 0
)
, N− =
(
0 1
1 0
)
,
and the Neumann–Zagier matrices of the mutation loop γ are given by
A+ =
(
2 0
0 2
)
, A− =
(
2 −1
−1 2
)
.
The following lemma is proved in [27].
Lemma 5.6. The matrix A+A
T
− is symmetric, where T is a transpose of a matrix.
Proof. This follows from Proposition 3.11 in [27]. �
In other words, Neumann–Zagier matrices satisfy the following relation:
A+A
T
− = A−A
T
+.
Exponents Associated with Y -Systems and their Relationship with q-Series 27
5.2 Definition of partition q-series
Let γ be a mutation loop. Let E and ∆ be the set of the black vertices and of the square vertices
of the mutation network Nγ , respectively.
We define two abelian groups Zγ and Bγ by
Zγ =
{
(u, v) |u ∈ Z∆, v ∈ Q∆, A−u = A+v
}
, Bγ =
{(
AT
+w,A
T
−w
)
|w ∈ ZE
}
.
It follows from Lemma 5.6 that Bγ is a subgroup of Zγ . Let Hγ be the quotient group of Zγ
by Bγ :
Hγ = Zγ/Bγ .
For any elements σ ∈ Hγ , we define σ≥0 ⊂ σ as follows:
σ≥0 = {(u, v) ∈ σ |ui ≥ 0 for all i = 1, . . . , N}.
For vectors u ∈ (Z≥0)∆ and v ∈ Q∆, we define a q-series W (x, y) by
W (u, v) =
q
1
2
u·v
(q)u1 · · · (q)uT
,
where u ·v is the usual dot product, that is, u ·v :=
T∑
i=1
uivi, and (q)n for a non-negative integer n
is defined by
(q)n =
n∏
k=1
(
1− qk
)
.
A mutation loop is called nondegenerate if A+ is a non-singular matrix. In addition, a non-
degenerate mutation loop is called positive definite if A−1
+ A− is a positive definite symmet-
ric matrix. Note that A−1
+ A− is always symmetric (if the mutation is nondegenerate) since
A−1
+ A− = A−1
+ A−A
T
+
(
A−1
+
)T
, and A−A
T
+ is symmetric.
Definition 5.7. Let γ be a nondegenerate positive definite mutation loop, and σ be an element
of Hγ . The partition q-series of γ associated with σ ∈ Hγ is defined by
Zσγ (q) =
∑
(u,v)∈σ≥0
W (u, v).
We also define the total partition q-series of γ as the sum of the partition q-series over Hγ :
Zγ(q) =
∑
σ∈Hγ
Zσγ (q).
Lemma 5.8. Suppose that γ is a nondegenerate positive definite mutation loop. Then the total
partition q-series is given by
Zγ(q) =
∑
u∈(Z≥0)∆
q
1
2
uTA−1
+ A−u
(q)u1 · · · (q)uT
.
28 Y. Mizuno
Proof. Because γ is nondegenerate, the abelian group Zγ can be written as
Zγ =
{(
u,A−1
+ A−u
)
|u ∈ Z∆
}
. (5.2)
This implies that
Zγ(q) =
∑
u∈(Z≥0)∆
W
(
u,A−1
+ A−u
)
=
∑
u∈(Z≥0)∆
q
1
2
uTA−1
+ A−u
(q)u1 · · · (q)uT
,
and the sum is well-defined since A−1
+ A− is positive definite. �
Corollary 5.9. Suppose that γ is a nondegenerate positive definite mutation loop. Then parti-
tion q-series of γ associated with any σ ∈ Hγ and the total partition q-series are well-defined as
a formal power series with integer coefficients in some rational power of q.
5.3 Asymptotics of partition q-series
In this section, we investigate the asymptotics of total partition q-series of nondegenerate positive
definite mutation loops. The asymptotics will be described by using the Rogers dilogarithm
function and the Jacobian matrix associated with the mutation loop.
Let γ be a nondegenerate positive definite mutation loop. For any t = 1, . . . , T , we define
rational functions z+(y; t) and z−(y; t) by
z+(y; t) =
Ymt(t− 1)
1 + Ymt(t− 1)
, z−(y; t) =
1
1 + Ymt(t− 1)
,
where Ymt(t− 1) is the rational function in y defined by (2.1).
Lemma 5.10. There is η ∈ (R>0)n such that µγ(η) = η, and such η is unique.
Proof. We define the following two sets:
F>0
γ =
{
η ∈ (R>0)n | fγ(η) = η
}
,
D(0,1)
γ =
{
ζ ∈ (0, 1)T |
T∏
t=1
ζ
−Aet+
t (1− ζt)A
et
− = 1 for all e ∈ E
}
.
Using [27, Corollary 3.9], we find the map Φ: F>0
γ → D
(0,1)
γ defined by
Φ(η) = z+(η; t)
is a bijection.
Since the matrix A+ is invertible, the set D
(0,1)
γ can be expressed as
D(0,1)
γ =
{
ζ ∈ (0, 1)T | ζs =
T∏
t=1
(1− ζt)(A−1
+ A−)st for all s = 1, . . . , T
}
. (5.3)
Using the fact that A−1
+ A− is positive definite and [36, Lemma 2.1], we find that the right-hand
side of (5.3) consists of one element. Thus F>0
γ also consists of one element. �
Exponents Associated with Y -Systems and their Relationship with q-Series 29
The Rogers dilogarithm function L(x) is defined by
L(x) = −1
2
∫ x
0
{
log(1− y)
y
+
log y
1− y
}
, 0 ≤ x ≤ 1.
Using the Rogers dilogarithm function, we define a positive real number a by
a =
T∑
t=1
L(z+(η; t)),
where η is the unique solution of the fixed point equation µγ(y) = y as in Lemma 5.10.
Let J(y) be the Jacobian matrix of the rational function µγ :
Jγ(y) :=
(
∂µi
∂yj
(y)
)
i,j∈I
,
where µi is the i-th components of µγ .
Let ε be a positive real number, and ε→ 0 denote the limit in the positive real axis.
Theorem 5.11. Let γ be a nondegenerate positive definite mutation loop. Then
lim
ε→0
Zγ
(
e−ε
)
e−
a
ε =
√
detA+
det(I − Jγ(η))
. (5.4)
Proof. Using [36, Theorem 2.3], we find that
lim
ε→0
Zγ
(
e−ε
)
e−
a′
ε = b′,
where a′ and b′ are the real numbers given by
a′ =
T∑
t=1
(
L(1)− L(z−(η; t))
)
,
b′ = det
(
A−1
+ A− + diag(ξ1, . . . , ξT )
)− 1
2
T∏
t=1
z+(η; t)−
1
2 ,
and ξt = z−(η; t)/z+(η; t). Since Rogers dilogarithm satisfies L(x) + L(1 − x) = L(1), we find
that a′ = a.
Let Z± = diag(z±(η; 1), . . . , z±(η;T )). Then we compute the number b as follows:
b′ = det
(
A−1
+ A− + diag(ξ1, . . . , ξT )
)− 1
2
T∏
t=1
z+(η; t)−
1
2
= det
(
A−1
+ A− + diag(ξ1, . . . , ξT )
)− 1
2 (detZ+)−
1
2
= det
(
A−1
+ A−Z+ + Z−
)− 1
2 = (detA+)
1
2 det(A+Z− +A−Z+)−
1
2 .
Using [27, Theorem 4.2], we find that
det(A+Z− +A−Z+) = det(I − Jγ(η)),
completing the proof. �
Remark 5.12. In [36] (as well as [35, 39]), they not only gave the limit of q-series, but also gave
a formula on the asymptotic expansion in the powers of ε. Although it might be interesting to
investigate higher order terms of the asymptotic expansion, it is not dealt with in this paper.
30 Y. Mizuno
1 2 r − 1
r
r + 12r − 22r − 1
· · ·
· · ·
A2r−1 :
Br :
1 2
· · ·
r − 1 r
1 2 r − 1
r
r + 1
· · ·Dr+1 :
Cr :
1 2
· · ·
r − 1 r
Figure 8. The map τ for Xr = Br, Cr.
5.4 Partition q-series of the mutation loop on Q(Xr, `)
Let Xr be a finite type Dynkin diagram, and ` be a positive integer such that ` ≥ 2. Let
γ = γ(Xr, `) be the mutation loop defined in Section 3.3. In this section, we give an explicit
expression of the partition q-series of γ(Xr, `).
For a Dynkin diagram of type Xr, we define another Dynkin diagram of type Yr′ as follows:
Yr′ =
Xr if X = A,D or E,
A2r−1 if Xr = Br,
Dr+1 if Xr = Cr,
E6 if Xr = F4,
D4 if Xr = G2.
We define a map
τ : {1, . . . , r′} → {1, . . . , r}
between the nodes of these two Dynkin diagrams by
τ(a) = a
if X = A,D or E,
τ(a) =
{
a if 1 ≤ a ≤ r,
2r − a if r + 1 ≤ a ≤ 2r − 1
if Xr = Br,
τ(a) =
{
a if 1 ≤ a ≤ r,
r if a = r + 1
Exponents Associated with Y -Systems and their Relationship with q-Series 31
1 2
3 4
56
E6 :
F4 :
1 2 3 4
1
23
4
D4 :
G2 :
1 2
Figure 9. The map τ for Xr = F4, G2.
if Xr = Cr,
τ(a) =
a if 1 ≤ a ≤ 4,
2 if a = 5,
1 if a = 6
if Xr = F4, and
τ(a) =
{
1 if a = 1, 3 or 4,
2 if a = 2
if Xr = G2. These definitions are indicated in Figs. 8 and 9.
We define integers κa (a = 1, . . . , r′) by
κa = #{b | 1 ≤ b ≤ r′, τ(a) = τ(b)}.
Then the vertices in the quiver Q(Xr, `) can be parameterized by the elements of the set
I = {(a,m) | 1 ≤ a ≤ r′, 1 ≤ m ≤ κa`− 1}.
Here we use the letter I so that we identify the set of the vertices in the quiver with this set.1
Let J be the set defined by
J = {(a,m) | 1 ≤ a ≤ r, 1 ≤ m ≤ ta`− 1}.
Lemma 5.13. The map
τ ′ : I −→ J
defined by τ ′(a,m) = (τ(a),m) is well-defined and surjective.
Proof. The lemma follows from the fact that κa = tb if τ(a) = b. �
Lemma 5.14. The induced map
τ̃ ′ : I/ν −→ J
defined by τ̃ ′([(a,m)]) = τ ′(a,m) is well-defined and bijective.
1This parametrization is slightly different from that in Section 3.3 that we used to define a mutation loop
on Q(G2, `).
32 Y. Mizuno
Proof. From the definitions of ν and τ ′, we see that (a,m) and (b, k) belong to the same ν-orbit
if and only if τ ′(a,m) = τ ′(b, k). Thus τ̃ ′ is well-defined and injective. It is also surjective from
Lemma 5.13. �
Using Lemmas 3.4, 5.1 and 5.14, we identify E and ∆ with J. In particular, we think that
the Neumann–Zagier matrices A+ and A− are J× J matrices.
Let Aab,mk± be the ((a,m), (b, k))-entries of the Neumann–Zagier matrices A±. Let K be the
J` × J` matrix defined by
Kab,mk =
(
min(tbm, tak)− mk
`
)
(αa |αb),
where Kab,mk is the ((a,m), (b, k))-entry of K. Let C̄a =
(
C̄amk
)
m,k=1,...,ta`−1
be the Cartan
matrix of type Ata`−1:
C̄amk = 2δmk − δm,k−1 − δm,k+1.
Proposition 5.15. The Neumann–Zagier matrices of γ(Xr, `) are given by
Aab,mk+ = δabC̄
a
mk,
Aab,mk− =
ta`−1∑
n=1
C̄amnKab,nk.
Proof. First, we note that the components of A+ and A− come from the arrows of the vertical
and horizontal directions, respectively, in the figures of quivers in Section 3.2. Then, the formula
for A+ that we want follows from the definition of γ(Xr, `) because the components of A+ come
from arrows in the vertical Ata`−1 quivers. We can also verify from the definition of γ(Xr, `)
that the following formula for A− holds:
Aab,mk− =
−Cab(δ2m,k−1 + 2δ2m,k + δ2m,k+1) if tb/ta = 2,
−Cab(δ3m,k−2 + 2δ3m,k−1 + 3δ3m,k + 2δ3m,k+1 + δ3m,k+2) if tb/ta = 3,
−Cbaδtbm,tak otherwise,
where Cab is the (a, b)-entry of the Cartan matrix of type Xr:
Cab =
2(αa |αb)
(αa |αa)
.
By concrete calculation, we can see that the right-hand side of this equation coincides with
ta`−1∑
n=1
C̄amnKab,nk,
and this complete the proof. �
Corollary 5.16. The following relation holds:
K = A−1
+ A−.
Moreover, γ = γ(Xr, `) is nondegenerate and positive definite, and µγ(y) = y has a unique
positive real solution.
Proof. The equation follows from Proposition 5.15. The positive definiteness ofK is well-known
(e.g., see the proof of Proposition 1.8 in [12]). The last argument follows from Lemma 5.10. �
Exponents Associated with Y -Systems and their Relationship with q-Series 33
Let Q be the free abelian group defined by
Q =
r⊕
a=1
Zαa.
The free abelian group Q is called a root lattice. Let M be the free abelian subgroup of Q
defined by
M =
r⊕
a=1
Ztaαa.
For any element u ∈ ZJ, let u
(a)
m denote the (a,m)-entry of u.
Lemma 5.17. The group homomorphism
F̄ : Zγ −→ Q
defined by
F̄ (u, v) =
∑
(a,m)∈J
mu(a)
m αa
is surjective, and the kernel of ι ◦ F̄ is equal to Bγ, where ι : Q→ Q/`M is the projection. Thus
we obtain the group isomorphism
F : Hγ −→ Q/`M. (5.5)
Proof. Since A+ is non-singular, the abelian group Zγ is given by (5.2). Thus the map F̄ is
surjective since
F̄ (u, v) =
r∑
a=1
u
(a)
1 αa
if u
(a)
m = 0 for all m > 1.
For any element w ∈ ZJ, we compute
F̄
(
AT
+w,A
T
−w
)
=
∑
(a,m)∈J
m
∑
(b,k)∈J
δabC̄
a
mkw
(b)
k
αa
=
r∑
a=1
(
ta`−1∑
m=1
2mw(a)
m −
ta`−2∑
m=1
(m+ 1)w(a)
m −
ta`−1∑
m=2
(m− 1)w(a)
m
)
αa
=
r∑
a=1
ta`w
(a)
ta`−1αa,
and this shows that Bγ = ker
(
ι ◦ F̄
)
as desired. �
Theorem 5.18. The partition q-series of γ = γ(Xr, `) associated with σ ∈ Hγ is given by
Zσγ (q) =
∑
u∈(Z≥0)J,(3)
q
1
2
uTKu∏
(a,m)∈J
(q)
u
(a)
m
,
where the sum runs over u ∈ (Z≥0)J under the condition∑
(a,m)∈J
mu(a)
m αa ≡ λ mod `M, (3)
where λ is a representative of F (σ).
34 Y. Mizuno
Proof. The theorem follows from Corollary 5.16 and Lemma 5.17. �
Corollary 5.19. The total partition q-series of γ = γ(Xr, `) is given by
Zγ(q) =
∑
u∈(Z≥0)J
q
1
2
uTKu∏
(a,m)∈J
(q)
u
(a)
m
.
Example 5.20. Let (Xr, `) = (A3, 3). The quiver Q(A3, 3) is given by
(1, 1)
−
(1, 2)
+
(2, 1)
+
(2, 2)
−
(3, 1)
−
(3, 2)
+
The set of indices of Q(A3, 3) is
I = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)}.
The mutation loop γ = γ(A3, 3) is given by
−
+
+
−
−
+
µ+−−→
−
+
+
−
−
+
µ−−−→
−
+
+
−
−
+
The mutation network Nγ is given by
(1, 1)
(1, 2)
(2, 1)
(2, 2)
(3, 1)
(3, 2)
(1, 1)
(1, 2)
(2, 1)
(2, 2)
(3, 1)
(3, 2)
Exponents Associated with Y -Systems and their Relationship with q-Series 35
The set of indices of Nγ is
J = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)}. (5.6)
The adjacency matrices of Nγ are given by N0 = 2I and
N+ =
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
, N− =
0 0 1 0 0 0
0 0 0 1 0 0
1 0 0 0 1 0
0 1 0 0 0 1
0 0 1 0 0 0
0 0 0 1 0 0
,
so the Neumann–Zagier matrices of γ are given by
A+ =
2 −1 0 0 0 0
−1 2 0 0 0 0
0 0 2 −1 0 0
0 0 −1 2 0 0
0 0 0 0 2 −1
0 0 0 0 −1 2
, A− =
2 0 −1 0 0 0
0 2 0 −1 0 0
−1 0 2 0 −1 0
0 −1 0 2 0 −1
0 0 −1 0 2 0
0 0 0 −1 0 2
,
where we choose an order of J as in (5.6).
Lemma 5.6 implies that
A+A
T
− = A−A
T
+ =
4 −2 −2 1 0 0
−2 4 1 −2 0 0
−2 1 4 −2 −2 1
1 −2 −2 4 1 −2
0 0 −2 1 4 −2
0 0 1 −2 −2 4
is a symmetric matrix. When X = A,D or E, the Neumann–Zagier matrices A+ and A−
themselves are symmetric, so we find that they are commuting, that is, A+A− = A−A+. Fur-
thermore, they are (possibly decomposable) symmetric Cartan matrices. We remark that such
a pair of commuting Cartan matrices appeared in the study of W -graphs by Stembridge [31],
and also in the classification of periodic mutations by Galashin and Pylyavskyy [7].
The positive definite matrix A−1
+ A− is given by
A−1
+ A− =
1
3
4 2 −2 −1 0 0
2 4 −1 −2 0 0
−2 −1 4 2 −2 −1
−1 −2 2 4 −1 −2
0 0 −2 −1 4 2
0 0 −1 −2 2 4
.
The partition q-series are parametrized by elements of the following set:
Hγ = {(a, 0, b, 0, c, 0) | 0 ≤ a, b, c,≤ 2},
where (u, v)+Zγ ∈ Hγ is denoted by
(
u
(1)
1 , u
(2)
1 , u
(3)
1 , u
(3)
2 , u
(3)
3
)
. Let us write (a, 0, b, 0, c, 0) more
simply as abc. Then we have
Hγ = {000, 100, 200, 010, 110, 210, 020, 120, 220, 001, 101, 201, 011, 111, 211, 021, 121, 221,
002, 102, 202, 012, 112, 212, 022, 122, 222}.
36 Y. Mizuno
We exhibit several low order terms of the partition q-series:
Zσγ (q) = 1 + 6q2 + 20q3 + 54q4 + 144q5 + 360q6 + 804q7 +O
(
q8
)
if σ = 000,
q
2
3 + 3q
5
3 + 13q
8
3 + 38q
11
3 + 108q
14
3 + 264q
17
3 + 622q
20
3 + 1364q
23
3 +O
(
q
26
3
)
if σ = 100, 200, 010, 110, 020, 220, 001, 011, 111, 002, 022, 222,
q + 6q2 + 18q3 + 56q4 + 144q5 + 357q6 + 808q7 + 1767q8 +O
(
q9
)
if σ = 210, 120, 211, 021, 221, 012, 112, 122, and
2q
4
3 + 8q
7
3 + 28q
10
3 + 76q
13
3 + 199q
16
3 + 468q
19
3 + 1060q
22
3 + 2256q
25
3 +O
(
q
28
3
)
if σ = 101, 201, 121, 102, 202, 212.
Example 5.21. Let (Xr, `) = (B3, 2). The quiver Q(B3, 2) is given by
(1, 1)
+
(5, 1)
−
(2, 1)
−
(3, 1)
+
(3, 2)
−
(3, 3)
+
(4, 1)
+
The set of indices of Q(B3, 2) is
I = {(1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 1), (5, 1)}.
The mutation loop γ = γ(B3, 2) is given by
+ −−
+
−
+
+
µ+µ+−−−→ + −−
+
−
+
+
µ−−−→ + −−
+
−
+
+
where the first quiver and the last quiver are identified by the left-right reflection ν.
Exponents Associated with Y -Systems and their Relationship with q-Series 37
The mutation network Nγ is given by
(1, 1) (2, 1)
(3, 3)
(3, 2)
(3, 1)
(1, 1) (2, 1)
(3, 3)
(3, 2)
(3, 1)
The set of indices of Nγ is
J = {(1, 1), (2, 1), (3, 1), (3, 2), (3, 3)}. (5.7)
The adjacency matrices of Nγ are given by N0 = 2I and
N+ =
0 0 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 0 1 0
, N− =
0 1 0 0 0
1 0 1 2 1
0 0 0 0 0
0 1 0 0 0
0 0 0 0 0
,
so the Neumann–Zagier matrices of γ are given by
A+ =
2 0 0 0 0
0 2 0 0 0
0 0 2 −1 0
0 0 −1 2 −1
0 0 0 −1 2
, A− =
2 −1 0 0 0
−1 2 −1 −2 −1
0 0 2 0 0
0 −1 0 2 0
0 0 0 0 2
,
where we choose an order of J as in (5.7).
The matrix A− is not symmetric, but
A+A
T
− = A−A
T
+ =
4 −2 0 0 0
−2 4 0 −2 0
0 0 4 −2 0
0 −2 −2 4 −2
0 0 0 −2 4
is symmetric as promised by Lemma 5.6.
The positive definite matrix A−1
+ A− is given by
A−1
+ A− =
1
2
2 −1 0 0 0
−1 2 −1 −2 −1
0 −1 3 2 1
0 −2 2 4 2
0 −1 1 2 3
.
38 Y. Mizuno
The partition q-series are parametrized by elements of the following set:
Hγ = {(a, b, c, 0, 0) | 0 ≤ a, b ≤ 1, 0 ≤ c ≤ 3},
where (u, v) + Zγ ∈ Hγ is denoted by
(
u
(1)
1 , u
(2)
1 , u
(3)
1 , u
(3)
2 , u
(3)
3
)
. Let us write (a, b, c, 0, 0) more
simply as abc. Then we have
Hγ = {000, 100, 010, 110, 001, 101, 011, 111, 002, 102, 012, 112, 003, 103, 013, 113}.
We exhibit several low order terms of the partition q-series:
Zσγ (q) = 1 + 9q2 + 21q3 + 66q4 + 144q5 + 349q6 + 723q7 +O
(
q8
)
if σ = 000,
q
1
2 + 4q
3
2 + 13q
5
2 + 38q
7
2 + 97q
9
2 + 228q
11
2 + 504q
13
2 + 1057q
15
2 +O
(
q
17
2
)
if σ = 100, 010, 110, 102, 012, 112,
q
3
4 + 5q
7
4 + 17q
11
4 + 48q
15
4 + 120q
19
4 + 279q
23
4 + 608q
27
4 + 1261q
31
4 +O
(
q
35
4
)
if σ = 001, 011, 111, 003, 013, 113,
3q
5
4 + 9q
9
4 + 30q
13
4 + 75q
17
4 + 187q
21
4 + 411q
25
4 + 885q
29
4 + 1783q
33
4 +O
(
q
37
4
)
if σ = 101, 103, and
3q + 6q2 + 25q3 + 57q4 + 156q5 + 334q6 + 744q7 + 1491q8 +O
(
q9
)
if σ = 002.
6 Relationships with affine Lie algebras
6.1 Affine Lie algebras
In this section, we review basic concepts of affine Lie algebras and their integrable highest weight
modules. See [14] for more detail. Let Xr (= Ar, Br, Cr, Dr, E6,7,8, F4 or G2) be a finite type
Dynkin diagram, and g be the finite dimensional simple Lie algebra of type Xr over C. Let h be
a Cartan subalgebra of g, and ∆ be the set of roots. We also use the notations on root systems
that we used in Section 3.1. We extend the inner product (· | ·) to the nondegenerate symmetric
bilinear form on h∗. We define the following two free abelian groups:
Q =
r⊕
a=1
Zαa, M =
r⊕
a=1
Ztaαa.
We also define the following sets:
P =
{
Λ ∈ h∗
∣∣∣∣ 2(Λ |αa)
(αa |αa)
∈ Z for all a = 1, . . . , r
}
,
P+ =
{
Λ ∈ P
∣∣∣∣ 2(Λ |αa)
(αa |αa)
≥ 0 for all a = 1, . . . , r
}
.
Let
ĝ = g⊗ C
[
t, t−1
]
⊕ CK ⊕ Cd
Exponents Associated with Y -Systems and their Relationship with q-Series 39
be the affine Lie algebra associated with g. The Lie bracket on ĝ is defined as[
X ⊗ tm, Y ⊗ tn
]
= [X,Y ]⊗ tm+n +mδm+n,0 · (X |Y )K,
[K, ĝ] = {0},[
d,X ⊗ tn
]
= nX ⊗ tn.
Fix a non-negative integer `. Let us define the following set:
P `+ = {Λ ∈ P+ | (Λ | θ) ≤ `},
where θ is the highest root in ∆. For any Λ ∈ P `+, there is the unique level ` integrable highest
weight ĝ-module such that the classical part of its highest weight is Λ, which is denoted by L(Λ).
We define the following rational numbers associated with L(Λ):
c(`) =
`dim g
`+ h∨
, hΛ =
(Λ |Λ + 2ρ)
2(`+ h∨)
.
Let q = e2πiτ . For any diagonalizable linear map α : V → V with eigenvalues λ1, λ2, . . . with
finite multiplicities m1,m2, . . . , we define the trace trV q
α by trV q
α =
∑
imiq
λi . We define the
following two functions as in [16]:
χΛ(τ) = q−
c(`)
24 trL(Λ) q
hΛ−d, bΛλ (τ) = q−
c(`)−r
24 trU(Λ,λ) q
hΛ− (λ |λ)
2`
−d, (6.1)
where λ ∈ h∗ and
U(Λ, λ) =
{
v ∈ L(Λ) |
(
h⊗ tn
)
v = δn,0λ(h)v for all h ∈ h, n ≥ 0
}
.
These series converge to holomorphic functions on the upper half-plane H = {τ ∈ C | Im τ > 0}.
The function χΛ(τ) is called the (specialized) normalized character2 of L(Λ), and bΛλ (τ) is called
the branching function for the pair (g, h). We will simply call bΛλ (τ) as the branching function.
Note that dividing this branching function by the r-th power of the Dedekind eta function yields
the string function in [15].
The asymptotics of the normalized character and the branching function were studied in
detailed in [15, 16]. Let τ ↓ 0 denote the limit in the positive imaginary axis.
Theorem 6.1 ([15, 16]). Suppose that λ ∈ Λ +Q. Then
lim
τ↓0
χΛ(τ)e−
πic(`)
12τ = a(Λ), (6.2)
lim
τ↓0
bΛλ (τ)e−
πi(c(`)−r)
12τ = |P/Q|
1
2 |Q/`M |−
1
2a(Λ), (6.3)
where a(Λ) is the real number defined by
a(Λ) =
∣∣P/(`+ h∨
)
M
∣∣− 1
2
∏
α∈∆+
2 sin
π(Λ + ρ |α)
`+ h∨
.
The real number a(Λ) is called the asymptotic dimension of L(Λ) for the following reason. The
module L(Λ) is infinite dimensional unless Λ is trivial, and χΛ(0) = dimL(Λ) = ∞. However,
(6.2) says that by multiplying by the appropriate exponential term and taking the limit, we
can get the finite number a(Λ). Therefore, we can think that the real number a(Λ) is the
“dimension” of L(Λ).
2More precisely, χΛ(τ) is the normalized character in [16] at z = t = 0.
40 Y. Mizuno
6.2 Exponents and asymptotic dimension
The branching functions with Λ = 0 are expected to coincide with the partition q-series that
we studied in Section 5.4 via the conjectural formula of the branching functions in [22]. This
was first observed in [17] for the total partition q-series with X = ADE. Let bΛλ (q) denote the
formal q-series defined by the right-hand side of (6.1).
Conjecture 6.2 ([22]). Suppose that ` ≥ 2. Let γ = γ(Xr, `) be the mutation loop defined in
Section 3.3. Suppose that σ ∈ Hγ, and λ ∈ Q is a representative of F (σ), where F is defined
as (5.5). Then
q−
c(`)−r
24 Zσγ (q) = b0λ(q). (6.4)
By summing (6.4) for all elements in Q/`M , we obtain the following conjectural identity on
the total partition q-series:
q−
c(`)−r
24 Zγ(q) =
∑
λ∈Q/`M
b0λ(q). (6.5)
By comparing the asymptotics (5.4) and (6.3), we find that (6.5) yields the following identities:
6
π2
T∑
t=1
L
(
z+(η; t)
)
= c(`)− r, (6.6)
and
1√
I − Jγ(η)
= |P/Q|
1
2a(0), (6.7)
where we use detA+ = |Q/`M |, which follows from Proposition 5.15.
The identity (6.6) is called the dilogarithm identity in conformal field theories, and proved
in [12, 13, 29] by using cluster algebras. On the other hand, the identity (6.7) is exactly Con-
jecture 3.8 at x = 1 because
NXr,`(1)
DXr,`(1)
=
r∏
a=1
ta
(
`+ h∨
) ∏
α∈∆+
2 sin
π(ρ |α)
`+ h∨
−2
=
∣∣Q/(`+ h∨
)
M
∣∣ ∏
α∈∆+
2 sin
π(ρ |α)
`+ h∨
−2
= |P/Q|−1a(0)−2.
Consequently, the conjectural identity (6.7) gives us a consistency between our conjecture on
exponents (Conjecture 3.8) and the known conjecture on q-series (Conjecture 6.2), and also gives
an interesting connection between the theory of cluster algebras and the representation theory
of affine Lie algebras.
Acknowledgements
The author gratefully acknowledges the help provided by Yuji Terashima. This work is supported
by JSPS KAKENHI Grant Number JP18J22576.
Exponents Associated with Y -Systems and their Relationship with q-Series 41
References
[1] Bartlett N., Warnaar S.O., Hall–Littlewood polynomials and characters of affine Lie algebras, Adv. Math.
285 (2015), 1066–1105, arXiv:1304.1602.
[2] Calegari F., Garoufalidis S., Zagier D., Bloch groups, algebraic K-theory, units, and Nahm’s conjecture,
arXiv:1712.04887.
[3] Flohr M., Grabow C., Koehn M., Fermionic expressions for the characters of cp,1 logarithmic conformal field
theories, Nuclear Phys. B 768 (2007), 263–276, arXiv:hep-th/0611241.
[4] Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529,
arXiv:math.RT/0104151.
[5] Fomin S., Zelevinsky A., Y -systems and generalized associahedra, Ann. of Math. 158 (2003), 977–1018,
arXiv:hep-th/0111053.
[6] Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112–164,
arXiv:math.RA/0602259.
[7] Galashin P., Pylyavskyy P., The classification of Zamolodchikov periodic quivers, Amer. J. Math. 141
(2019), 447–484, arXiv:1603.03942.
[8] Garoufalidis S., Lê T.T.Q., Nahm sums, stability and the colored Jones polynomial, Res. Math. Sci. 2
(2015), Art. 1, 55 pages, arXiv:1112.3905.
[9] Georgiev G., Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic
space, arXiv:q-alg/9504024.
[10] Gleitz A.-S., On the KNS conjecture in type E, Ann. Comb. 18 (2014), 617–643, arXiv:1307.2738.
[11] Hatayama G., Kuniba A., Okado M., Takagi T., Tsuboi Z., Paths, Crystals and Fermionic Formulae, in
MathPhys odyssey, 2001, Prog. Math. Phys., Vol. 23, Birkhäuser Boston, Boston, MA, 2002, 205–272,
arXiv:math.QA/0102113.
[12] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T-systems and Y-systems, diloga-
rithm identities, and cluster algebras I: type Br, Publ. Res. Inst. Math. Sci. 49 (2013), 1–42, arXiv:1001.1880.
[13] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T-systems and Y-systems, diloga-
rithm identities, and cluster algebras II: types Cr, F4, and G2, Publ. Res. Inst. Math. Sci. 49 (2013), 43–85,
arXiv:1001.1881.
[14] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
[15] Kac V.G., Peterson D.H., Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math.
53 (1984), 125–264.
[16] Kac V.G., Wakimoto M., Modular and conformal invariance constraints in representation theory of affine
algebras, Adv. Math. 70 (1988), 156–236.
[17] Kato A., Terashima Y., Quiver mutation loops and partition q-series, Comm. Math. Phys. 336 (2015),
811–830, arXiv:1403.6569.
[18] Keller B., The periodicity conjecture for pairs of Dynkin diagrams, Ann. of Math. 177 (2013), 111–170,
arXiv:1001.1531.
[19] Kirillov A.N., Identities for the Rogers dilogarithmic function connected with simple Lie algebras, J. Soviet
Math. 47 (1989), 2450–2459.
[20] Kuniba A., Thermodynamics of the Uq
(
X
(1)
r
)
Bethe ansatz system with q a root of unity, Nuclear Phys. B
389 (1993), 209–244.
[21] Kuniba A., Nakanishi T., Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys.
Lett. A 7 (1992), 3487–3494, arXiv:hep-th/9206034.
[22] Kuniba A., Nakanishi T., Suzuki J., Characters in conformal field theories from thermodynamic Bethe
ansatz, Modern Phys. Lett. A 8 (1993), 1649–1659, arXiv:hep-th/9301018.
[23] Kuniba A., Nakanishi T., Suzuki J., T -systems and Y -systems in integrable systems, J. Phys. A: Math.
Theor. 44 (2011), 103001, 146 pages, arXiv:1010.1344.
[24] Lee C.-H., Nahm’s conjecture and Y -systems, Commun. Number Theory Phys. 7 (2013), 1–14,
arXiv:1109.3667.
[25] Lee C.-H., A proof of the KNS conjecture: Dr case, J. Phys. A: Math. Theor. 46 (2013), 165201, 12 pages,
arXiv:1210.1669.
https://doi.org/10.1016/j.aim.2015.08.011
https://arxiv.org/abs/1304.1602
https://arxiv.org/abs/1712.04887
https://doi.org/10.1016/j.nuclphysb.2007.01.025
https://arxiv.org/abs/hep-th/0611241
https://doi.org/10.1090/S0894-0347-01-00385-X
https://arxiv.org/abs/math.RT/0104151
https://doi.org/10.4007/annals.2003.158.977
https://arxiv.org/abs/hep-th/0111053
https://doi.org/10.1112/S0010437X06002521
https://arxiv.org/abs/math.RA/0602259
https://doi.org/10.1353/ajm.2019.0010
https://arxiv.org/abs/1603.03942
https://doi.org/10.1186/2197-9847-2-1
https://arxiv.org/abs/1112.3905
https://arxiv.org/abs/q-alg/9504024
https://doi.org/10.1007/s00026-014-0244-7
https://arxiv.org/abs/1307.2738
https://doi.org/10.1007/978-1-4612-0087-1_9
https://arxiv.org/abs/math.QA/0102113
https://doi.org/10.4171/PRIMS/95
https://arxiv.org/abs/1001.1880
https://doi.org/10.4171/PRIMS/96
https://arxiv.org/abs/1001.1881
http://dx.doi.org/10.1017/CBO9780511626234
https://doi.org/10.1016/0001-8708(84)90032-X
https://doi.org/10.1016/0001-8708(88)90055-2
https://doi.org/10.1007/s00220-014-2224-5
https://arxiv.org/abs/1403.6569
https://doi.org/10.4007/annals.2013.177.1.3
https://arxiv.org/abs/1001.1531
https://doi.org/10.1007/BF01840426
https://doi.org/10.1007/BF01840426
https://doi.org/10.1016/0550-3213(93)90291-V
https://doi.org/10.1142/S0217732392002895
https://doi.org/10.1142/S0217732392002895
https://arxiv.org/abs/hep-th/9206034
https://doi.org/10.1142/S0217732393001392
https://arxiv.org/abs/hep-th/9301018
https://doi.org/10.1088/1751-8113/44/10/103001
https://doi.org/10.1088/1751-8113/44/10/103001
https://arxiv.org/abs/1010.1344
https://doi.org/10.4310/CNTP.2013.v7.n1.a1
https://arxiv.org/abs/1109.3667
https://doi.org/10.1088/1751-8113/46/16/165201
https://arxiv.org/abs/1210.1669
42 Y. Mizuno
[26] Lee C.-H., Positivity and periodicity of Q-systems in the WZW fusion ring, Adv. Math. 311 (2017), 532–568,
arXiv:1302.1467.
[27] Mizuno Y., Jacobian matrices of Y-seed mutations, Adv. in Appl. Math. 115 (2020), 101987, 36 pages,
arXiv:1805.00044.
[28] Nahm W., Recknagel A., Terhoeven M., Dilogarithm identities in conformal field theory, Modern Phys.
Lett. A 8 (1993), 1835–1847, arXiv:hep-th/9211034.
[29] Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case,
Nagoya Math. J. 202 (2011), 23–43, arXiv:0909.5480.
[30] Nakanishi T., Periodicities in cluster algebras and dilogarithm identities, in Representations of Algebras and
Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 407–443, arXiv:1006.0632.
[31] Stembridge J.R., Admissible W -graphs and commuting Cartan matrices, Adv. in Appl. Math. 44 (2010),
203–224.
[32] Stoyanovsky A.V., Lie algebra deformations and character formulas, Funct. Anal. Appl. 32 (1998), 66–68.
[33] Stoyanovsky A.V., Feigin B.L., Functional models for representations of current algebras and semi-infinite
Schubert cells, Funct. Anal. Appl. 28 (1994), 55–72.
[34] Terashima Y., Yamazaki M., N = 2 theories from cluster algebras, Progr. Theor. Exper. Phys. 2014 (2012),
023B01, 37 pages, arXiv:1301.5902.
[35] Terhoeven M., Dilogarithm identities, fusion rules and structure constants of CFTs, Modern Phys. Lett. A
9 (1994), 133–141, arXiv:hep-th/9307056.
[36] Vlasenko M., Zwegers S., Nahm’s conjecture: asymptotic computations and counterexamples, Commun.
Number Theory Phys. 5 (2011), 617–642, arXiv:1104.4008.
[37] Warnaar S.O., Proof of the Flohr–Grabow–Koehn conjectures for characters of logarithmic conformal field
theory, J. Phys. A: Math. Theor. 40 (2007), 12243–12254, arXiv:0704.3118.
[38] Warnaar S.O., Zudilin W., Dedekind’s η-function and Rogers–Ramanujan identities, Bull. Lond. Math. Soc.
44 (2012), 1–11, arXiv:1001.1571.
[39] Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry. II, Springer,
Berlin, 2007, 3–65.
https://doi.org/10.1016/j.aim.2017.02.031
https://arxiv.org/abs/1302.1467
https://doi.org/10.1016/j.aam.2019.101987
https://arxiv.org/abs/1805.00044
https://doi.org/10.1142/S0217732393001562
https://doi.org/10.1142/S0217732393001562
https://arxiv.org/abs/hep-th/9211034
https://doi.org/10.1215/00277630-1260432
https://arxiv.org/abs/0909.5480
https://doi.org/10.4171/101-1/9
https://arxiv.org/abs/1006.0632
https://doi.org/10.1016/j.aam.2009.08.001
https://doi.org/10.1007/BF02465764
https://doi.org/10.1007/BF01079010
https://doi.org/10.1093/ptep/ptt115
https://arxiv.org/abs/1301.5902
https://doi.org/10.1142/S0217732394000149
https://arxiv.org/abs/hep-th/9307056
https://doi.org/10.4310/CNTP.2011.v5.n3.a2
https://doi.org/10.4310/CNTP.2011.v5.n3.a2
https://arxiv.org/abs/1104.4008
https://doi.org/10.1088/1751-8113/40/40/015
https://arxiv.org/abs/0704.3118
https://doi.org/10.1112/blms/bdr019
https://arxiv.org/abs/1001.1571
https://doi.org/10.1007/978-3-540-30308-4_1
1 Introduction
2 Quiver mutations and Y-seed mutations
2.1 Quiver mutations
2.2 Y-seed mutations
3 Exponents determined from a pair (Xr, )
3.1 Root systems
3.2 Quiver Q(Xr,)
3.3 Mutation loop on Q(Xr,)
3.4 Periodicity
3.5 Exponents of Q(Xr, )
3.6 Conjecture on exponents
3.7 Examples
4 Proofs of Theorem 3.9
4.1 (A1, ) case
4.2 (Ar, 2) case
5 Partition q-series
5.1 Mutation networks
5.2 Definition of partition q-series
5.3 Asymptotics of partition q-series
5.4 Partition q-series of the mutation loop on Q(Xr, )
6 Relationships with affine Lie algebras
6.1 Affine Lie algebras
6.2 Exponents and asymptotic dimension
References
|
| id | nasplib_isofts_kiev_ua-123456789-210582 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:16Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mizuno, Yuma 2025-12-12T10:29:11Z 2020 Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages 1815-0659 2020 Mathematics Subject Classification: 13F60;17B22;81R10 arXiv:1812.05863 https://nasplib.isofts.kiev.ua/handle/123456789/210582 https://doi.org/10.3842/SIGMA.2020.028 Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the formulation of the -system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type ᵣ, and prove this conjecture for (A₁,ℓ) and (Aᵣ,2) cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from q-series identities for this relationship. The author gratefully acknowledges the help provided by Yuji Terashima. This work is supported by JSPS KAKENHI Grant Number JP18J22576. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exponents Associated with Y-Systems and their Relationship with q-Series Article published earlier |
| spellingShingle | Exponents Associated with Y-Systems and their Relationship with q-Series Mizuno, Yuma |
| title | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_full | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_fullStr | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_full_unstemmed | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_short | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_sort | exponents associated with y-systems and their relationship with q-series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210582 |
| work_keys_str_mv | AT mizunoyuma exponentsassociatedwithysystemsandtheirrelationshipwithqseries |