1 Introduction
The representations of a quiver, introduced by Gabriel [Reference Gabriel7] in the 1970s, emerge as a notion of significant interest due to a remarkable relation with Lie theory. Gabriel revealed a bijective correspondence between the set of dimension vectors of indecomposable representations of a Dynkin quiver and the root system of the corresponding semisimple Lie algebra. Kac [Reference Kac9] extended Gabriel’s theorem to an arbitrary quiver.
Kac’s work shows the validity of the geometric approach to classification of indecomposable representations of quivers. Consider the representation variety
$R(Q,\alpha )$
associated to a fixed dimension vector
$\alpha $
of a quiver Q, on which there is an algebraic group action. Then the isoclasses of representations are identified with the orbits under the action. Let
$V_p$
be the representation corresponding to a point p in
$R(Q,\alpha )$
. According to the Krull–Schmidt theorem,
$V_p\cong \oplus V_{i,p}$
for some indecomposable representations
$V_{i,p}$
. Kac proved that the set of dimension vectors
$\{\mathbf {dim}\, V_{i,p}\}$
is independent of the choice of p in some dense open subset
${\mathcal U}$
in
$R(Q,\alpha )$
. This leads to the notion of a canonical decomposition
$\alpha =\sum \alpha _i$
(with
$\mathbf {dim}\, V_{i,p}=\alpha _i$
) and enables us to study quiver representations by a recursive method (see [Reference Derksen and Weyman5, Reference Schofield11]).
Let
${\mathbb F}_q$
be the finite field with q elements. Let
$M_Q(\alpha ,q)$
be the number of isoclasses of representations of a quiver Q over
${\mathbb F}_q$
with a fixed dimension vector
$\alpha $
. This polynomial is closely related to Kac’s conjecture (see [Reference Crawley-Boevey and Van den Bergh3, Reference Hua8–Reference Kac10]).
We investigate the relationships between the two families of polynomials
$M_Q(\alpha ,q)$
and
$\prod _{i} M_Q(\alpha _i,q)$
associated with the canonical decomposition
$\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$
when Q is a tame quiver of type
$\tilde {A}_n$
. This allows to collect evidence relevant to questions proposed in [Reference Chen and Liu2, Conjecture 0.1]. The main tools are basic homological properties of module categories, especially Auslander–Reiten theory.
2 Preliminaries
In this section, we give a brief review on quiver representations and the canonical decomposition (for more details, see [Reference Auslander, Reiten and Smalø1, Reference Deng, Du, Parshall and Wang4]).
Let
${\mathbb F}$
be a field. Let
$Q=(Q_0,Q_1)$
be a quiver with vertex set
$Q_0$
and arrow set
$Q_1$
. We always assume that the quiver is finite, that is,
$Q_0$
and
$Q_1$
are finite sets, and write
$Q_0 = \{1, 2, \ldots , n\}$
. The starting and terminating vertices of an arrow
$\alpha $
will be denoted by
$t(\alpha )$
and
$h(\alpha )$
respectively. By definition, a finite dimensional representation
$V=(V_i,V_{\alpha })$
of Q consists of a set of finite dimensional vector spaces
$V_i$
for all
$i\in Q_0$
and of a set of linear transformations
$V_\alpha : V_{t(\alpha )}\rightarrow V_{h(\alpha )}$
for all
$\alpha \in Q_1$
. A representation is said to be indecomposable if it cannot be written as the direct sum of two nonzero representations. The dimension vector
$\mathbf {dim}\, V$
associated with a representation V is a nonnegative integer-valued function on the vertex set
$Q_0$
and so an element of
${\mathbb Z} Q_0$
. We identify
${\mathbb Z} Q_0$
with
${\mathbb Z}^n$
in what follows.
For each
$\mu =(\,\mu _i)\in {\mathbb Z}^n$
, let
$\mathop{\mathrm{supp}}\nolimits \mu =\{i\in Q_0\mid \mu _i\not =0\}$
be the support of
$\mu $
. We say that
$\mathop{\mathrm{supp}}\nolimits \mu $
is connected if the full subquiver of Q with vertex set
$\mathop{\mathrm{supp}}\nolimits \mu $
is connected.
Now assume that
${\mathbb F}$
is an algebraically closed field. Fix a dimension vector
$\alpha $
of Q. The representation variety associated with
$\alpha $
is the affine space
$$ \begin{align*}R(Q,\alpha):=\prod\limits_{\rho:i\rightarrow j}{\mathrm{Hom}\,}({\mathbb F}^{\alpha_i},{\mathbb F}^{\alpha_j}).\end{align*} $$
Thus, each element x in
$R(Q,\alpha )$
determines a representation
$V_x$
of Q with dimension vector
$\alpha $
. The algebraic group
$GL_{\alpha }({\mathbb F}):=\prod _{i\in Q_0}GL_{\alpha _i}({\mathbb F})$
acts on
$R(Q,\alpha )$
by conjugation. It is known that
$GL_{\alpha }({\mathbb F})$
-orbits
${\mathcal O}_x$
correspond bijectively to isoclasses
$[V_x]$
of representations of Q with dimension vector
$\alpha $
.
Let Q be a quiver with n vertices and
$\alpha =(\alpha _i),\beta =(\,\beta _i)\in {\mathbb Z}^n$
. The bilinear forms
$$ \begin{align*}\langle \alpha,\beta\,\rangle:=\sum_{i=1}^n\alpha_i\beta_i-\sum_{\alpha\in Q_1}\alpha_{t(\alpha)}\,\beta_{h(\alpha)}\quad\mbox{and}\quad (\alpha,\beta):=\langle \alpha,\beta\,\rangle+\langle\,\beta,\alpha\rangle\end{align*} $$
on
${\mathbb Z}^n$
are called the Euler form and the symmetric Euler form of Q, respectively.
We now recall from [Reference Kac9, Reference Kac10] the notion of canonical decomposition of a dimension vector. Generic representations of
$\alpha $
are said to have the property P if there exists a dense open subset
${\mathcal U}$
of
$R(Q,\alpha )$
such that all representations parametrised by points in
${\mathcal U}$
have the property P.
Definition 2.1. The canonical decomposition of a dimension vector
$\alpha $
is the sum
$\alpha =\beta _1+\beta _2+\cdots +\beta _s$
provided that generic representations of
$\alpha $
have indecomposable summands whose dimension vectors are given by these
$\beta _i$
.
By definition, the canonical decomposition of a fixed dimension vector is unique up to reordering.
Definition 2.2. Write
$\beta \hookrightarrow \alpha $
for the property that there exists a dense open subset
${\mathcal U}$
such that each representation parametrised by points in
${\mathcal U}$
of dimension vector
$\alpha $
has a subrepresentation of dimension vector
$\beta $
.
Given two dimension vectors
$\alpha ,\beta $
of Q, denote by
${\mathrm {hom}\,}(\alpha ,\beta )$
and
${\mathrm {ext}\,}(\alpha ,\beta )$
the minimal value of the upper semi-continuous functions
$\mathrm {dim}\,{\mathrm {Hom}\,}_Q(-,-)$
and
$\mathrm {dim}\,{\mathrm {Ext}\,}^1_Q(-,-))$
on
$R(Q,\alpha )\times R(Q,\beta )$
, respectively.
Definition 2.3. We call
$\alpha $
a Schur root if
$\alpha $
is the dimension vector of a representation with endormorphism ring
${\mathbb F}$
.
Theorem 2.4 [Reference Kac10]
The sum
$\alpha =\beta _1+\beta _2+\cdots +\beta _s$
is the canonical decomposition if and only if
$\beta _i$
is a Schur root and
${\mathrm {ext}\,}(\,\beta _i,\beta _j)=0$
for
$i\not =j$
.
Theorem 2.5 [Reference Schofield11]
We have
${\mathrm {ext}\,}(\alpha ,\beta )=\max _{\alpha ^{\prime }\hookrightarrow \alpha }\{-\langle \alpha ^{\prime },\beta \,\rangle \}.$
3 Main results
Throughout this section, assume that the quiver Q of type
$\tilde {A}_n$
, where
$n=s+t$
, has s arrows going clockwise and t arrows going anticlockwise:
We first recall some properties of module categories of tame quivers of type
$\tilde {A}_n$
from [Reference Dlab and Ringel6]. Note that representations of Q are identified with left modules over the path algebra A of Q. Denote by
$\Gamma (Q)$
the Auslander–Reiten quiver of A. It is known that
$\Gamma (Q)$
has one preprojective component, one preinjective component and infinitely many regular tubes including two nonhomogeneous tubes
${\mathcal T}_1$
and
${\mathcal T}_2$
with ranks t and s, respectively. We say M is a preprojective, preinjective or regular module according as all indecomposable summands of M belong to the preprojective, preinjective or regular component of
$\Gamma (Q)$
, respectively.
In what follows, we define by convention the partial order
$<$
on the set of roots of Q by saying
$\alpha <\beta $
if each component of
$\alpha $
is less than or equal to that of
$\beta $
. Note that
$\delta =(1,1,\ldots ,1)\in {\mathbb Z}^n$
is the minimal positive imaginary root of Q.
Definition 3.1. Let Q be a quiver of type
$\tilde {A_n}$
. Let
$\alpha =n\delta +\gamma $
with
$n\in {\mathbb N}$
and a real root
$\gamma $
satisfying
$0<\gamma <\delta $
.
-
(1) If
$\{b_j\}_{j=0}^t\subseteq \mathop{\mathrm{supp}}\nolimits \gamma $
, then we call
$\alpha $
a lower arc. If
$\{a_i\}_{i=0}^s\subseteq \mathop{\mathrm{supp}}\nolimits \gamma $
, then we call
$\alpha $
an upper arc. -
(2) If
$\mathop{\mathrm{supp}}\nolimits \gamma =\{a_i\}_{i=i_0}^s \cup \{b_j\}_{j=j_0}^t \; (1\leq i_0\leq s,1\leq j_0\leq t)$
, then we call
$\alpha $
a right arc. If
$\mathop{\mathrm{supp}}\nolimits \gamma =\{a_i\}_{i=0}^{s_0} \cup \{b_j\}_{j=0}^{t_0} \;(0\leq s_0\leq s-1,1\leq t_0\leq t-1)$
, then we call
$\alpha $
a left arc. -
(3) If
$\mathop{\mathrm{supp}}\nolimits \gamma =\{a_i\}_{i=u}^v \;(0<u<v<s)$
or
$\mathop{\mathrm{supp}}\nolimits \gamma =\{b_j\}_{j=u'}^{v'} \;(0<u'<v'<t)$
, then we call
$\alpha $
a short arc.
Remark 3.2. An indecomposable
$kQ$
-module M is preprojective or preinjective if and only if
$\mathbf {dim}\, M$
is a right or a left arc, respectively. If M is an indecomposable
$kQ$
-module with
$\mathbf {dim}\, M$
an upper or a lower arc, then M is regular. Moreover,
$\mathbf {dim}\, M$
is an upper or a lower arc if and only if M lies in
${\mathcal T}_1$
or
${\mathcal T}_2$
, respectively.
Example 3.3. Let Q be the quiver of type
$\tilde {A}_5$
.
Then
$\alpha _1=(1,1,0,1,1,1)$
is a lower arc and
$\alpha _2=(3,3,3,2,2,3)$
is an upper arc;
$\alpha _3=(3,4,4,3,4,4)$
is a right arc and
$\alpha _4=(1,1,0,1,0,0)$
is a left arc;
$\alpha _5=(0,1,1,0,0,0)$
and
$\alpha _6=(1,1,1,2,2,1)$
are both short arcs.
For canonical decompositions of quivers of type
$\tilde {A}_n$
, we follow Schofield’s inductive algorithm [Reference Schofield11]. For more details, see [Reference Chen and Liu2, Examples 2.8 and 2.9], which is our case when
$n=3$
and
$(s,t)=(2,1)$
.
Lemma 3.4. Let
$\alpha $
,
$\beta $
be real roots of Q satisfying
$0<\alpha ,\,\beta <\delta $
and
$\alpha +\beta \geq \delta $
. Then
${\mathrm {ext}\,}(\alpha ,\beta )=0={\mathrm {ext}\,}(\,\beta ,\alpha )$
if and only if
$\alpha $
and
$\beta $
are a lower arc and an upper arc, respectively.
Proof. Suppose that
$\alpha $
and
$\beta $
are a lower arc and an upper arc, respectively. From Remark 3.2, there exist indecomposable regular modules
$V,W$
with dimension vectors
$\alpha ,\beta $
, which lie in
${\mathcal T}_2$
and
${\mathcal T}_1$
, respectively. It follows that
$$ \begin{align*}\mathrm{dim}\, {\mathrm{Ext}\,}_Q^1(V,W)=\mathrm{dim}\, {\mathrm{Ext}\,}_Q^1(W,V)=0.\end{align*} $$
Therefore,
${\mathrm {ext}\,}(\alpha ,\beta )=0={\mathrm {ext}\,}(\,\beta ,\alpha )$
.
Conversely, suppose that there are two real roots
$\alpha , \beta <\delta $
satisfying
$\alpha +\beta \geq \delta $
and
${\mathrm {ext}\,}(\alpha ,\beta )=0={\mathrm {ext}\,}(\,\beta ,\alpha )$
. We claim that
$\alpha +\beta>\delta $
. Indeed if
$\alpha +\beta =\delta $
, then
$$ \begin{align*}\begin{array}{lll} (\alpha,\beta)&\!\!\!\!=(\alpha,\delta-\alpha)=-(\alpha,\alpha)=-2\\ &\!\!\!\!=\langle\alpha,\beta\,\rangle+\langle\,\beta,\alpha\rangle. \end{array}\end{align*} $$
Thus,
$\langle \alpha ,\beta \,\rangle <0$
(or
$\langle \,\beta ,\alpha \rangle <0$
), that is,
${\mathrm {ext}\,}(\alpha ,\beta )>0$
(or
${\mathrm {ext}\,}(\,\beta ,\alpha )>0$
), which is a contradiction.
If
$\alpha $
is the dimension vector of an indecomposable preprojective module, that is,
$\alpha $
is a right arc, put
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits \alpha=\{a_i\}_{i=i_0}^s \cup \{b_j\}_{j=j_0}^t \quad (1\leq i_0\leq s,1\leq j_0\leq t).\end{align*} $$
Case 1:
$\beta $
is the dimension vector of an indecomposable preprojective module. Clearly,
$\alpha +\beta \ngeq \delta $
, which is a contradiction.
Case 2:
$\beta $
is the dimension vector of an indecomposable preinjective module. Assume that
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits \beta=\{a_i\}_{i=0}^{s_0} \cup \{b_j\}_{j=0}^{t_0} \quad (0\leq s_0\leq s-1,1\leq t_0\leq t-1).\end{align*} $$
Thus,
$s_0\geq i_0-1$
and
$t_0\geq j_0-1$
. Consequently,
$$ \begin{align*}\langle\,\beta,\alpha\rangle=s_0-i_0+1+t_0-j_0+1-(s_0-(i_0-1)+1+t_0-(\,j_0-1)+1)=-2,\end{align*} $$
which implies that
${\mathrm {ext}\,}(\,\beta ,\alpha )>0$
, which is a contradiction.
Case 3:
$\beta $
is the dimension vector of an indecomposable regular module. Since
${\alpha +\beta>\delta} $
and
$\beta <\delta $
, it follows that
$\beta $
is a lower arc or an upper arc. If
$\beta $
is a lower arc, then
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits\beta=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^m \cup \{a_i\}_{i=n}^s \quad (0\leq m<n\leq s)\end{align*} $$
and
$m\geq i_0-1$
. Consequently,
$$ \begin{align*} \langle\,\beta,\alpha\rangle & =m-i_0+1+s-n+1+t-1-j_0+1 \\ &\quad -(m-(i_0-1)+1+s-1-n+1+t-1-(\,j_0-1)+1) \\ & =-1. \end{align*} $$
This implies that
${\mathrm {ext}\,}(\,\beta ,\alpha )>0$
, contrary to the hypothesis. If
$\beta $
is an upper arc, a similar argument also leads to a contradiction.
Case 4:
$\alpha $
is the dimension vector of an indecomposable preinjective module. This case can be handled similarly and leads to a contradiction.
Therefore,
$\alpha $
and
$\beta $
are both dimension vectors of regular modules. Since
$\alpha +\beta>\delta $
, it follows that there exists at least one dimension vector which is a lower arc or an upper arc. Assume that
$\alpha $
is a lower arc, that is,
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits\alpha=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^m \cup \{a_i\}_{i=n}^s \quad (0\leq m<n\leq s).\end{align*} $$
We claim that
$\beta $
is an upper arc. Indeed, on the one hand, if
$\beta $
is a short arc, that is,
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits \beta=\{a_i\}_{i=u}^v \quad (0<u<v<s),\end{align*} $$
then
$v\geq n-1$
and
$m\geq u-1$
. Let
$\beta ^{\prime }$
be a dimension vector satisfying that
$0<\beta ^{\prime }<\delta $
and
$\mathop{\mathrm{supp}}\nolimits \beta ^{\prime }=\{a_i\}_{i=n-1}^v.$
Thus,
${\mathrm {ext}\,}(\,\beta ^{\prime },\beta -\beta ^{\prime })=0.$
Therefore,
$\beta ^{\prime }\,{\hookrightarrow}\, \beta $
and
$$ \begin{align*}\langle\,\beta^{\prime},\alpha\rangle=v-n+1-(v-(n-1)+1)=-1,\end{align*} $$
which implies that
${\mathrm {ext}\,}(\,\beta ,\alpha )>0$
, which is a contradiction. On the other hand, if
$\beta $
is a lower arc, that is,
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits\beta=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^{m^{\prime}} \cup \{a_i\}_{i={n}^{\prime}}^s \quad (0\leq m^{\prime}<n^{\prime}\leq s),\end{align*} $$
then
$m\geq n^{\prime }-1$
or
$m^{\prime }\geq n-1$
. Since
$\alpha +\beta \geq \delta $
and
$\alpha <\delta $
, it follows that
$\beta>\delta $
, which is a contradiction. In conclusion,
$\alpha $
and
$\beta $
are a lower arc and an upper arc, respectively.
Lemma 3.5. Assume that
$\alpha $
,
$\beta $
are real roots of Q such that
$0<\alpha ,\beta <\delta $
and
$\alpha +\beta \geq \delta $
. If
${\mathrm {ext}\,}(\alpha ,\beta )=0={\mathrm {ext}\,}(\,\beta ,\alpha )$
, then
$$ \begin{align*}{\mathrm{ext}\,}(\delta,\alpha)=0={\mathrm{ext}\,}(\alpha,\delta)\quad \text{and}\quad {\mathrm{ext}\,}(\delta,\beta)=0={\mathrm{ext}\,}(\,\beta,\delta).\end{align*} $$
Proof. According to Lemma 3.4,
$\alpha ,\;\beta $
are a lower arc and an upper arc, respectively. Suppose
$\alpha $
is a lower arc. Set
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits \alpha=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=1}^m \cup \{a_i\}_{i=n}^{s} \quad (0\leq m<n\leq s).\end{align*} $$
It is straightforward to calculate that
$$ \begin{align*} \langle\alpha,\delta\rangle =m+s-n+t+1-(m+1+t+s-n)=0. \end{align*} $$
Applying the fact that
$(\alpha ,\delta )=0$
, we get
$\langle \delta ,\alpha \rangle =0$
. Let
$X_{\alpha }$
be the unique indecomposable module of dimension vector
$\alpha $
. Put
$$ \begin{align*} Y_{\delta}=N\oplus\bigoplus_{j=m+2}^{n-1}S_{a_j} \quad\mbox{and}\quad Y_{\delta}^{\prime}=N'\oplus \bigoplus_{j=m+1}^{n-2}S_{a_j} \end{align*} $$
for indecomposable modules
$N,N'$
with
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits \mathbf{dim}\, N=\mathop{\mathrm{supp}}\nolimits \alpha\cup \{a_{m+1}\}\quad\text{and}\quad \mathop{\mathrm{supp}}\nolimits \mathbf{dim}\, N'=\mathop{\mathrm{supp}}\nolimits \alpha\cup \{a_{n-1}\},\end{align*} $$
respectively. It is easy to check that
$$ \begin{align*}{\mathrm{Hom}\,}(X_{\alpha},N)=0\quad \text{and}\quad {\mathrm{Hom}\,}(N',X_{\alpha})=0.\end{align*} $$
Hence,
${\mathrm {Hom}\,}(X_{\alpha },Y_{\delta })=0$
and
${\mathrm {Hom}\,}(Y^{\prime }_{\delta },X_{\alpha })=0$
. It follows that
$$ \begin{align*}\mathrm{dim}\,{\mathrm{Ext}\,}(X_{\alpha},Y_{\delta})=0\;\quad\text{and}\quad \mathrm{dim}\,{\mathrm{Ext}\,}(Y^{\prime}_{\delta},X_{\alpha})=0,\end{align*} $$
which implies that
${\mathrm {ext}\,}(\alpha ,\delta )=0={\mathrm {ext}\,}(\delta ,\alpha ).$
The other case is treated similarly.
Lemma 3.6. Let
$\alpha _1=n_1\delta +\gamma _1$
and
$\alpha _2=n_2\delta +\gamma _2$
be two dimension vectors, where
$\gamma _1,\gamma _2$
are real roots satisfying
$0<\gamma _1,\gamma _2<\delta $
, and
$\gamma _1+\gamma _2\geq \delta $
. Suppose
${\mathrm {ext}\,}(\alpha _1,\alpha _2)=0={\mathrm {ext}\,}(\alpha _2,\alpha _1)$
. Then
${\mathrm {ext}\,}(\gamma _1,\gamma _2)=0={\mathrm {ext}\,}(\gamma _2,\gamma _1)$
.
Proof. Suppose on the contrary that
${\mathrm {ext}\,}(\gamma _1,\gamma _2)\neq 0$
or
${\mathrm {ext}\,}(\gamma _2,\gamma _1)\neq 0$
.
Case 1:
$\gamma _1$
is a dimension vector of an indecomposable preprojective module.
(1) If
$\gamma _2$
is a dimension vector of an indecomposable preprojective module, then we have
$\gamma _1+\gamma _2\ngeq \delta $
, which is a contradiction.
(2) If
$\gamma _2$
is a dimension vector of an indecomposable preinjective module, since there is no nonzero homomorphism from preinjective modules to preprojective modules, it follows that
$\langle \gamma _2,\gamma _1\rangle <0$
and, by a direct calculation,
$\langle \delta ,\gamma _1\rangle <0$
and
$\langle \gamma _2,\delta \rangle <0$
. Therefore,
$\langle \alpha _1,\alpha _2\rangle =\langle n_2\delta +\gamma _2,n_1\delta +\gamma _1\rangle =n_2\langle \delta ,\gamma _1\rangle +n_1\langle \gamma _2,\delta \rangle +\langle \gamma _2,\gamma _1\rangle <0$
, which forces
${\mathrm {ext}\,}(\alpha _2,\alpha _1)\neq 0$
, which is a contradiction.
(3) If
$\gamma _2$
is a dimension vector of an indecomposable regular module, then we have
${\langle \gamma _2,\gamma _1\rangle <0}$
and
$\langle \delta ,\gamma _2\rangle =0=\langle \gamma _2,\delta \rangle <0$
. This implies that
$$ \begin{align*}\langle n_2\delta+\gamma_2,n_1\delta+\gamma_1\rangle=n_2\langle\delta,\gamma_1\rangle+n_1\langle\gamma_2,\delta\rangle+\langle\gamma_2,\gamma_1\rangle<0,\end{align*} $$
so
${\mathrm {ext}\,}(\alpha _2,\alpha _1)\neq 0$
, which is a contradiction.
Case 2.
$\gamma _1$
is the dimension vector of an indecomposable preinjective module. This is the case dual to Case 1.
Case 3.
$\gamma _1$
,
$\gamma _2$
are the dimension vectors of indecomposable regular modules. Since
$\gamma _1+\gamma _2>\delta $
, at least one of
$\gamma _1, \;\gamma _2$
is a lower arc or an upper arc. Assume that
$\gamma _1$
is a lower arc, that is,
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits\gamma_1=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^m \cup \{a_i\}_{i=n}^s \quad (0\leq m<n\leq s).\end{align*} $$
If
$\gamma _2$
is a short arc, we put
$\mathop{\mathrm{supp}}\nolimits \gamma _2=\{\alpha _i\}_{i=u}^v\;(0<u<v<s)$
. Thus,
$i_m\geq u-1, i_n\geq v-1$
. By Lemma 3.4, there exists
$\gamma ^{\prime }_2 \in {\mathbb N}^n$
such that
$0<\gamma ^{\prime }_2 <\delta $
and
$\gamma ^{\prime }_2 \hookrightarrow \gamma _2$
. Moreover,
$\langle \gamma ^{\prime }_2,\gamma _1\rangle <0 $
. Consequently,
$n_2\delta +\gamma ^{\prime }_2\hookrightarrow n_2\delta +\gamma _2$
, and
$$ \begin{align*}\langle n_2\delta+\gamma^{\prime}_2,n_1\delta+\gamma_1\rangle=\langle\gamma^{\prime}_2,\gamma_1\rangle<0.\end{align*} $$
This implies that
${\mathrm {ext}\,}(\alpha _2,\alpha _1)>0$
, which is contrary to the hypothesis.
If
$\gamma _2$
is a lower arc, put
$$ \begin{align*}\mathop{\mathrm{supp}}\nolimits\gamma_2=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^{m'} \cup \{a_i\}_{i={n}'}^s \quad (0\leq m'<n'\leq s),\end{align*} $$
so
$i_m\geq n'-1$
or
$m'\geq i_n-1$
.
(1) If
$i_m\geq n'-1$
, then by Lemma 3.4, there exists
$\gamma ^{\prime }_1\in {\mathbb N}^n$
such that
$0<\gamma ^{\prime }_1<\delta $
satisfying
$\gamma _1\hookrightarrow \gamma _1$
and
$\langle \gamma ^{\prime }_1,\gamma _2\rangle <0 $
. Consequently,
$n_1\delta +\gamma ^{\prime }_1\hookrightarrow n_1\delta +\gamma _1$
and
$$ \begin{align*}\langle n_1\delta+\gamma^{\prime}_1,n_2\delta+\gamma_2\rangle=\langle\gamma^{\prime}_1,\gamma_2\rangle<0.\end{align*} $$
Hence,
${\mathrm {ext}\,}(\alpha _1,\alpha _2)>0$
, which is a contradiction.
(2) If
$m'\geq i_n-1$
, then by Lemma 3.4 again, there exists
$\gamma ^{\prime }_2\in {\mathbb N}^n$
such that
$0<\gamma ^{\prime }_2<\delta $
satisfying
$\gamma ^{\prime }_2\hookrightarrow \gamma _2$
and
$\langle \gamma _2,\gamma _1\rangle <0 $
. Thus we obtain
$n_2\delta +\gamma ^{\prime }_2\hookrightarrow n_2\delta +\gamma _2$
and
$$ \begin{align*}\langle n_2\delta+\gamma^{\prime}_2,n_1\delta+\gamma_1\rangle=\langle\gamma^{\prime}_2,\gamma_1\rangle<0.\end{align*} $$
Hence,
${\mathrm {ext}\,}(\alpha _2,\alpha _1)>0$
, which is a contradiction.
If
$\gamma _2$
is an upper arc, it follows from Lemma 3.4 that
$$ \begin{align*}{\mathrm{ext}\,}(\alpha_1,\alpha_2)=0={\mathrm{ext}\,}(\alpha_2,\alpha_1). \end{align*} $$
This is a contradiction. We deal with the case where
$\gamma _1$
is an upper arc by a similar argument. In conclusion, we have
${\mathrm {ext}\,}(\gamma _1,\gamma _2)=0={\mathrm {ext}\,}(\gamma _2,\gamma _1),$
as desired.
Lemma 3.7. Suppose that
$\alpha =n\delta +\gamma $
is the dimension vector of an indecomposable preprojective or preinjective module, where
$\gamma $
is a real root satisfying
$0<\gamma <\delta $
. Then
${\mathrm {ext}\,}(\delta ,\alpha )>0$
or
${\mathrm {ext}\,}(\alpha ,\delta )>0$
, respectively.
Proof. Since
$\alpha $
is the dimension vector of an indecomposable preprojective module,
$\alpha $
is a right arc and so
$\gamma $
is a right arc. Assume that
$\mathop{\mathrm{supp}}\nolimits \gamma =\{a_i\}_{i=i_0}^s \cup \{b_j\}_{j=j_0}^t\; \text {for} \;1\leq i_0\leq s \;\text {and}\;1\leq j_0\leq t.$
Therefore,
$$ \begin{align*}\langle\delta,\alpha\rangle=\langle\delta,\gamma\rangle=s-i_0+1+t-j_0-(s-i_0+1+t-j_0+1)=-1,\end{align*} $$
which implies
${\mathrm {ext}\,}(\delta ,\alpha )>0$
. The case for an indecomposable preinjective module is handled similarly. This completes the proof.
Proposition 3.8. Let
$\alpha =\underbrace {\delta +\cdots +\delta }_{m}+\,\alpha _1+\cdots +\alpha _l$
be the canonical decomposition of a dimension vector
$\alpha $
of Q. Let
$\alpha _i=n_i\delta +\gamma _i$
with
$0<\gamma _i<\delta $
for each i. Denote by
$s_1,s_2$
the number of lower arcs and upper arcs, respectively, in the decomposition. Then
$$ \begin{align*}\gamma_1+\gamma_2+\cdots+\gamma_l\not \geq (\min\{s_1,s_2\}+1)\delta.\end{align*} $$
Proof. We may assume that
$s_1\geq s_2$
.
Case 1:
$s_1=0$
. If
$m>0$
, by Theorem 2.4 and Lemma 3.7, each
$\alpha _i$
is the dimension vector of a regular module. Then all
$\gamma _i$
are short arcs. Clearly,
$\gamma _1+\gamma _2+\cdots +\gamma _l\not \geq \delta .$
If
$m=0$
, then assume that
$\gamma _1,\ldots ,\gamma _{t_1}$
are right arcs,
$\gamma _{t_1+1},\ldots ,\gamma _{t_1+t_2}$
are left arcs and
$\gamma _{t_1+t_2+1},\ldots \gamma _{t_1+t_2+t_3}$
are short arcs. Note that
$l=t_1+t_2+t_3$
. First we claim that
$\gamma _1+\cdots +\gamma _{t_1+t_2}\not \geq \delta $
. Otherwise, there exist
$\gamma _f \;(1\leq f\leq t_1)$
and
$\gamma _g \;(t_1< g\leq t_1+t_2)$
with
$$ \begin{align*} \mathop{\mathrm{supp}}\nolimits\gamma_f&= \{a_i\}_{i=i_0}^s \cup \{b_j\}_{j=j_0}^t \quad (1\leq i_0\leq s,1\leq j_0\leq t)\\ \mathop{\mathrm{supp}}\nolimits\gamma_g&=\{a_i\}_{i=0}^{s_0} \cup \{b_j\}_{j=0}^{t_0} \quad (0\leq s_0\leq s-1,1\leq t_0\leq t-1) \end{align*} $$
satisfying
$s_0\geq i_0-1$
. Thus,
$\langle \gamma _g,\gamma _f\rangle <0$
. By Lemma 3.6,
${\mathrm {ext}\,}(\alpha _g,\alpha _f)>0$
, which is a contradiction. Further, we claim that
$\gamma _1+\cdots +\gamma _l\not \geq \delta $
. Suppose on the contrary that
$\gamma _1+\cdots +\gamma _{t_1+t_2}\not \geq \delta $
and
$\gamma _1+\cdots +\gamma _l\geq \delta $
. Then there exist roots
$\gamma _f \; (1\leq f\leq t_1)$
and
$\gamma _h \;(t_1+t_2< h\leq ~l)$
with
$$ \begin{align*} \mathop{\mathrm{supp}}\nolimits \gamma_f&= \{a_i\}_{i=i_0}^s \cup \{b_j\}_{j=j_0}^t \quad (1\leq i_0\leq s,1\leq j_0\leq t),\\ \mathop{\mathrm{supp}}\nolimits\gamma_h&=\{a_i\}_{i=u}^{v} \quad (0<u<v<s), \end{align*} $$
for
$v\geq i_0-1$
. Hence,
$\langle \gamma _h,\gamma _f\rangle <0$
. Again by Lemma 3.6,
${\mathrm {ext}\,}(\alpha _h,\alpha _f)>0$
, which is a contradiction.
Case 2:
$s_1\neq 0$
. Assume that
$\gamma _1,\gamma _2,\ldots ,\gamma _{s_1}$
are all lower arcs and
$\gamma _{s_1+1},\ldots ,\gamma _{s_1+s_2}$
are upper arcs (the case
$s_2=0$
is included). We claim that
$\gamma _1+\gamma _2+\cdots +\gamma _{s_1+s_2}\not \geq (\min \{s_1,s_2\}+1)\delta .$
Otherwise, there exist
$\gamma _c,\gamma _d$
such that
$\gamma _c+\gamma _d>\delta $
for
$1\leq c,d\leq s_1+s_2$
. By Lemmas 3.4 and 3.7,
${\mathrm {ext}\,}(\alpha _c,\alpha _d)>0$
or
${\mathrm {ext}\,}(\alpha _d,\alpha _c)>0$
. This leads to a contradiction. Further, we claim that
$\gamma _1+\gamma _2+\cdots +\gamma _l\not \geq (\min \{s_1,s_2\}+1)\delta .$
Suppose on the contrary that
$\gamma _1+\gamma _2+\cdots +\gamma _l\geq (\min \{s_1,s_2\}+1)\delta .$
Then there exist
$\gamma _c, \gamma _e$
for some
$1\leq c \leq s_1$
and
$s_1+s_2<e\leq l$
such that
$$ \begin{align*} \mathop{\mathrm{supp}}\nolimits \gamma_c&=\{b_j\}_{j=0}^t \cup \{a_i\}_{i=0}^{i_m} \cup \{a_i\}_{i=i_n}^s \quad (0\leq i_m<i_n\leq s),\\ \mathop{\mathrm{supp}}\nolimits \gamma_e&=\{a_i\}_{i=u}^v \quad (0<u<v<s), \end{align*} $$
for
$v\geq i_n-1$
. Let
$0<\gamma ^{\prime }_e<\delta $
and
$\mathop{\mathrm{supp}}\nolimits \gamma ^{\prime }_e=\{a_i\}_{i=i_n-1}^v$
. Thus,
$\gamma ^{\prime }_e\hookrightarrow \gamma _e$
and
$$ \begin{align*}\langle\gamma^{\prime}_e,\gamma_c\rangle=v-i_n+1-(v-(i_n-1)+1)=-1<0,\end{align*} $$
which implies
${\mathrm {ext}\,}(\gamma _e,\gamma _c)>0$
. By Lemma 3.6,
${\mathrm {ext}\,}(\alpha _{e},\alpha _{c})>0$
, which is a contradiction.
For the rest of this section, let
${\mathbb F}={\mathbb F}_q$
be the finite field with q elements. Let
$M_Q(\alpha ,q)$
be the number of isoclasses of representations of a quiver Q over
${\mathbb F}$
with a fixed dimension vector
$\alpha $
.
Lemma 3.9 [Reference Chen and Liu2]
Suppose that Q is a tame quiver and
$\alpha =n_0\delta +\gamma $
is the dimension vector of Q for some
$\gamma \geq 0$
and
$\gamma \not \geq \delta $
. Then
$\deg (M_{Q}(\alpha ,q))=n_0$
.
There is no general formula for the canonical decomposition of a fixed dimension vector of Q. To address the issue, we follow Schofield’s inductive algorithm [Reference Schofield11], as in our previous paper [Reference Chen and Liu2].
Theorem 3.10. Assume that
$\alpha =\underbrace {\delta +\cdots +\delta }_{m}+\,\alpha _1+\cdots +\alpha _l$
is the canonical decomposition of a dimension vector
$\alpha $
of Q, where
$\alpha _i \;(i=1,2,\ldots ,l)$
are all real Schur roots. Then
$$ \begin{align*}\lim_{q\rightarrow\infty}\frac{M_Q(\alpha,q)} {(M_Q(\delta,q))^{m}q^x\prod_{i=1}^l M_Q(\alpha_i,q)} =\text{constant},\end{align*} $$
where
$s_1,\;s_2$
are the numbers of lower arcs and upper arcs, respectively, and
$x=\min \{s_1,s_2\}$
.
Proof. Since Q is a quiver of type
$\tilde {A}_n$
and
$\alpha _i \;(i=1,2,\ldots ,l)$
are all real roots, it follows that
$\alpha _i=n_i\delta +\gamma _i $
, where
$\gamma _i $
are all real roots and
$0<\gamma _i<\delta $
for
$i\in \{1,\ldots ,l\}$
. Now assume that
$s_1\geq s_2$
. (The case for
$s_2\geq s_1$
can be handled similarly.) By Proposition 3.8,
$$ \begin{align*}\gamma_1+\gamma_2+\cdots+\gamma_l\not \geq (\min\{s_1,s_2\}+1)\delta.\end{align*} $$
Again by Lemma 3.9,
$$ \begin{align*}\deg (M_{Q}(\alpha_i,q))=n_i\quad\text{and}\quad\deg (M_{Q}(\alpha,q))=m+n_1+n_2+\cdots +n_l+\min\{s_1,s_2\}.\end{align*} $$
Therefore,
$\deg (M_Q(\alpha ,q))\kern1.3pt{=}\kern1.3pt \sum _{i=1}^l \deg (M_Q(\alpha _i,q))\kern1.3pt{+}\kern1.3pt m\kern1.3pt{+}\kern1.3pt x.$
This completes the proof.
Example 3.11. As an illustration, we consider Example 3.10 in [Reference Chen and Liu2]. Let Q be the quiver of type
$\tilde {A}_2$
:
Then the canonical decomposition of
$\alpha =(2,3,3)$
is
$\alpha =(0,1,0)+(2,2,3)=\alpha _1+\alpha _2$
. An elementary calculation yields
$$ \begin{align*}M_{Q}(\alpha,q)=2q^2+16q+72, \quad M_{Q}(\alpha_1,q)=1, \quad M_{Q}(\alpha_2,q)=q^2+9q+44.\end{align*} $$
Thus,
$$ \begin{align*}\lim_{q\rightarrow\infty}\frac{M_Q(\alpha,q)} {(M_Q(\alpha_1,q))M_Q(\alpha_2,q)} =2,\end{align*} $$
which is consistent with the main theorem.
