Representations of regular trees and invariants of AR-components for generalized Kronecker quivers

TL;DR

This paper studies the structure of regular components in the Auslander-Reiten quiver of generalized Kronecker algebras, revealing a close relationship between quasi-rank and a geometric distance measure, and constructing a classification bijection.

Contribution

It introduces a novel connection between quasi-rank and a geometric distance in regular components, and constructs a classification bijection for components based on this distance.

Findings

The quasi-rank of a regular component is closely linked to the distance between specific cones.

A bijection between the field and regular components with fixed distance is established.

Results provide new insights into the enumeration of regular components by quasi-rank.

Abstract

We investigate the generalized Kronecker algebra $\mathcal{K}_r = k\Gamma_r$ with $r \geq 3$ arrows. Given a regular component $\mathcal{C}$ of the Auslander-Reiten quiver of $\mathcal{K}_r$, we show that the quasi-rank $rk(\mathcal{C}) \in \mathbb{Z}_{\leq 1}$ can be described almost exactly as the distance $\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0$ between two non-intersecting cones in $\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.\] Utilizing covering theory, we construct for each $n \in \mathbb{N}_0$ a bijection $\varphi_n$ between the field $k$ and $\{ \mathcal{C} \mid \mathcal{C} \ \text{regular component}, \ \mathcal{W}(\mathcal{C}) = n \}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.

Figures

Equations

$$-\mathcal{W}(\mathcal{C})\le \mathrm{rk}(\mathcal{C})\le -\mathcal{W}(\mathcal{C})+3.$$

$$\mathrm{rk}(\mathcal{C}):=\min \{l\in \mathbb{Z}\mid \forall m\ge l:\mathrm{rad}(X,{\tau }^{m}X)\ne 0\},$$

$$-\mathcal{W}(\mathcal{C})\le \mathrm{rk}(\mathcal{C})\le -\mathcal{W}(\mathcal{C})+3.$$

$$(\ast ){\mathcal{W}}_C(\mathcal{E})={\mathcal{W}}_C(\mathcal{D})+\mathrm{ql}(X)-1.$$

$$x^{+}:=\{y\in Q_0\mid \exists \alpha :x\to y\}\text{and}{x}^{-}:=\{y\in Q_0\mid \exists \alpha :y\to x\}.$$

$$M_x$$

$$\underline{\dim }:\mathrm{mod}kQ\to {\mathbb{Z}}^{Q_0},M\mapsto ({\dim }_k{M}_x{)}_{x\in Q_0}.$$

$$⟨\underline{\dim }M,\underline{\dim }N⟩={\dim }_k\mathrm{Hom}(M,N)-{\dim }_k\mathrm{Ext}(M,N).$$

$$M=M[1]\to M[2]\to M[3]\to \cdots \to M[l]\to \cdots$$

$$\cdots (l)M\to \cdots \to (3)M\to (2)M\to (1)M=M,$$

$$\mathrm{rk}(\mathcal{C}):=\min \{l\in \mathbb{Z}\mid \forall m\ge l:\mathrm{rad}(X,{\tau }^{m}X)\ne 0\}.$$

$$0>2-r=1+(r-1{)}^2-r(r-1)=⟨\underline{\dim }{X}_{\beta },\underline{\dim }{X}_{\alpha }⟩={\dim }_k\mathrm{Hom}(X_{\beta },X_{\alpha })-{\dim }_k\mathrm{Ext}(X_{\beta },X_{\alpha }).$$

$${\tau }^{-1}M\to X_{\alpha }\to \tau X_{\beta }\to \tau N,\text{and}$$

$$0\ne \mathrm{Hom}(M,{\tau }^{2}N)=\mathrm{Hom}({\tau }^{-1}{W}_{\mathcal{C}},{\tau }^2\circ {\tau }^{-\mathcal{W}(\mathcal{C})}{W}_{\mathcal{C}})=\mathrm{Hom}(W_{\mathcal{C}},{\tau }^{-\mathcal{W}(\mathcal{C})+3}{W}_{\mathcal{C}}).$$

$${\gamma }_i^{-1}{\gamma }_i\sim {ϵ}_1\text{and}{\gamma }_i{\gamma }_i^{-1}\sim {ϵ}_2.$$

$$g.[v]=[v{w}^{-1}]\text{and}$$

$$g.([u]\to [{\gamma }_{i}u])=([u{w}^{-1}]\to [{\gamma }_{i}u{w}^{-1}]).$$

$$(M^g{)}_x:=M_{g.x}\text{and}{M}^g(\alpha ):=M(g.\alpha ).$$

$${\pi }_{\lambda }(M{)}_i:=\underset{\pi (y)=i}{⨁}{M}_y\text{and}$$

$${\pi }_{\lambda }(M)({\gamma }_i):=\underset{\pi (\beta )={\gamma }_i}{⨁}M(\beta ):{\pi }_{\lambda }(M{)}_1\to {\pi }_{\lambda }(M{)}_2\text{for}1\le i\le r.$$

$$g_i:=\underset{\pi (y)=i}{⨁}{f}_y:{\pi }_{\lambda }(M{)}_i\to {\pi }_{\lambda }(N{)}_i.$$

$$\mathrm{Inj}:=\{M\in \mathrm{rep}(C_r)\mid \forall \delta \in (C_r{)}_1:M(\delta )\text{is injective}\}\text{and}$$

$$\mathrm{Surj}:=\{M\in \mathrm{rep}(C_r)\mid \forall \delta \in (C_r{)}_1:M(\delta )\text{is surjective}\}.$$

$$0={\iota }_y\circ M(\alpha )(m_x)=\left[\underset{\pi (\beta )=\pi (\alpha )}{⨁}M(\beta )\right]\circ {\iota }_x(m_x)=(N({\gamma }_{\overline{\alpha }})\circ {\iota }_x)(m_x).$$

$$0=(f(m){)}_y=\sum _{\beta \in (C_r{)}_1,t(\beta )=y}{\alpha }_{\overline{\beta }}M(\beta )(m_{s(\beta )})={\alpha }_{1}M(\gamma )(m_x)+\sum _{\gamma \ne \beta \in (C_r{)}_1,t(\beta )=y}{\alpha }_{\overline{\beta }}M(\beta )(m_{s(\beta )}).$$

$$0=(f(m){)}_a=\sum _{\beta \in (C_r{)}_1,t(\beta )=a}{\alpha }_{\overline{\beta }}M(\beta )(m_{s(\beta )})={\alpha }_{2}M(\eta )(m_x)+\sum _{\eta \ne \beta \in (C_r{)}_1,t(\beta )=a}{\alpha }_{\overline{\beta }}M(\beta )(m_{s(\beta )}).$$

$$(X^i{)}_y:=\{\begin{array}{l}k,y\in \{x_0\}\cup x_0^{+}\setminus \{z_i\}\\ 0,\text{else}\end{array}$$

$$g_i=\underset{\pi (y)=i}{⨁}{f}_y:\underset{\pi (y)=i}{⨁}(({\tau }_{C_r}^n{X}^i{)}^g{)}_y\to \underset{\pi (y)=i}{⨁}{M}_y.$$

$$d^{-}(X):=\min \{l\in \mathbb{Z}\mid {\tau }^{-l}X\in \mathrm{Inj}\}\text{and}{d}^{+}(X):=\min \{l\in \mathbb{Z}\mid {\tau }^{l}X\in \mathrm{Surj}\}.$$

$${\mathcal{W}}_C(\mathcal{D})=d^{+}(X)+d^{-}(X)-\mathrm{ql}(X).$$

Full text

[TABLE]

Utilizing covering theory, we construct for each $n\in\mathbb{N}_{0}$ a bijection $\varphi_{n}$ between the field $k$ and $\{\mathcal{C}\mid\mathcal{C}\ \text{regular component},\ \mathcal{W}(\mathcal{C})=n\}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.

Partly supported by the D.F.G. priority program SPP 1388 “Darstellungstheorie”

00footnotetext: 2010 Mathematics Subject Classification: 16G20, 16G6000footnotetext: Keywords: Kronecker algebra, Auslander-Reiten theory, Covering theory

Introduction

Let $k$ be an algebraically closed field of arbitrary characteristic. The finite-dimensional algebra $\mathcal{K}_{r}$ is defined as the path algebra of the quiver $\Gamma_{r}$ with vertices $1,2$, $r\in\mathbb{N}$ arrows $\gamma_{1},\ldots,\gamma_{r}\colon 1\to 2$ and called the generalized Kronecker algebra. We denote by $\operatorname{mod}\mathcal{K}_{r}$ the category of finite-dimensional left-modules of $\mathcal{K}_{r}$.

It is well known that for $r\geq 3$ the hereditary algebra $\mathcal{K}_{r}$ is of wild representation type [Ker3, 1.3,1.5], every regular component in the Auslander-Reiten quiver of $\mathcal{K}_{r}$ is of type $\mathbb{Z}A_{\infty}$ [Ri3] and there is a bijection between the regular components and the ground field $k$ [Assem3, XVIII 1.8]. Therefore, the problem of completely understanding the category $\operatorname{mod}\mathcal{K}_{r}$ or all regular components is considered hopeless and it is desirable to find invariants which give more specific information about the regular components.

One important invariant $($for any wild hereditary algebra$)$, introduced in [Ker1], is the quasi-rank $\operatorname{rk}(\mathcal{C})\in\mathbb{Z}$ of a regular component $\mathcal{C}$. For a quasi-simple module $X$ in $\mathcal{C}$, $\operatorname{rk}(\mathcal{C})$ is defined as

[TABLE]

where $\operatorname{rad}(X,Y)$ is the space of non-invertible homomorphisms from $X$ to $Y$.

Another interesting invariant $\mathcal{W}(\mathcal{C})\in\mathbb{N}_{0}$ was recently introduced in [Wor1]. Motivated by the representation theory of group algebras of $p$-elementary abelian groups of characteristic $p>0$, the author defines the category $\operatorname{EKP}$ of modules with the equal kernels property and the category $\operatorname{EIP}$ of modules with the equal images property in the framework of $\mathcal{K}_{r}$. She shows the existence of uniquely determined quasi-simple modules $M_{\mathcal{C}}$ and $W_{\mathcal{C}}$ in $\mathcal{C}$ such that the cone $(M_{\mathcal{C}}\rightarrow)$ of all successors of $M_{\mathcal{C}}$ satisfies $(M_{\mathcal{C}}\rightarrow)=\operatorname{EKP}\cap\mathcal{C}$ and the cone $(\rightarrow W_{\mathcal{C}})$ of all predecessors of $W_{\mathcal{C}}$ satisfies $(\rightarrow W_{\mathcal{C}})=\operatorname{EIP}\cap\mathcal{C}$. The width $\mathcal{W}(\mathcal{C})$ of $\mathcal{C}$ is defined as the unique number $\mathcal{W}(\mathcal{C})\in\mathbb{N}_{0}$ such that $\tau^{\mathcal{W}(\mathcal{C})+1}M_{\mathcal{C}}=W_{\mathcal{C}}$, i.e. the distance between the two cones.

Utilizing homological descriptions of $\operatorname{EKP}$ and $\operatorname{EIP}$ from [Wor1] involving a family of elementary modules, we show that the two invariants $\operatorname{rk}(\mathcal{C})$ and $\mathcal{W}(\mathcal{C})$ are linked by the inequality

[TABLE]

Motivated by this connection, we construct for each $n\in\mathbb{N}$ a regular component $\mathcal{C}$ with $\mathcal{W}(\mathcal{C})=n$. In order to do so, we consider representations over the universal covering $C_{r}$ of $\Gamma_{r}$.

We define classes $\operatorname{Inj}$,$\operatorname{Surj}$ of representations over $C_{r}$ such that $M\in\operatorname{rep}(C_{r})$ is in $\operatorname{Inj}$ $($resp. $\operatorname{Surj})$ if and only if for each arrow $\delta\colon x\to y$ of $C_{r}$ the linear map $M(\delta)\colon M_{x}\to M_{y}$ is injective $($resp. surjective$)$.

Let $\pi_{\lambda}\colon\operatorname{rep}(C_{r})\to\operatorname{rep}(\Gamma_{r})$ be the push-down functor [Gab3, 2.7] and $M\in\operatorname{rep}(C_{r})$ be indecomposable. We prove that $M$ is in $\operatorname{Inj}$ $($resp. $\operatorname{Surj})$ if and only if $\pi_{\lambda}(M)$ is in $\operatorname{EKP}$ $($resp. $\operatorname{EIP})$. Since a component $\mathcal{D}$ of the Auslander-Reiten quiver of $C_{r}$ which is taken to a regular component $\mathcal{C}:=\pi_{\lambda}(\mathcal{D})$ is also of type $\mathbb{Z}A_{\infty}$, we can lift the definition of $\mathcal{W}(\mathcal{C})$ to $\mathcal{D}$. We define $\mathcal{W}_{C}(\mathcal{D})\in\mathbb{N}_{0}$ as the distance between the cones $\operatorname{Surj}\cap\mathcal{D}$ and $\operatorname{Inj}\cap\mathcal{D}$ and show that $\mathcal{W}_{C}(\mathcal{D})=\mathcal{W}(\mathcal{C})$.

For $X\in\mathcal{D}$, we denote by $\operatorname{ql}(X)$ its quasi-length. If $X$ has certain properties, we show how to construct a short exact sequence $\delta_{X}=0\to Y\to E\to X\to 0$ with indecomposable middle term $E$ in a component $\mathcal{E}$ such that

[TABLE]

The construction of $\delta_{X}$ relies on the fact that $C_{r}$ is an infinite $r$-regular tree with bipartite orientation. Using $(\ast)$, we construct for each $n\in\mathbb{N}$ a component $\mathcal{D}_{n}$ with $\mathcal{W}_{C}(\mathcal{D}_{n})=n$.

In conjunction with a natural action of $\operatorname{GL}_{r}(k)$ on $\operatorname{rep}(\Gamma_{r})$, we arrive at our main theorem:

Theorem**.**

Let $n\in\mathbb{N}_{0}$. There is a bijection $k\to\{\mathcal{C}\mid\mathcal{C}\ \text{regular component of}\ \Gamma_{r},\mathcal{W}(\mathcal{C})=n\}$.

As an immediate consequence we get the following statements, which are generalizations of results by Kerner and Lukas [KerLuk2, 3.1], [KerLuk2, 5.2] for the Kronecker algebra.

Corollary**.**

Let $r\geq 3$, then for each $n\in\mathbb{N}$ there are exactly $|k|$ regular components with quasi-rank in $\{-n,-n+1,-n+2,-n+3\}$.

Corollary**.**

Assume that $k$ is uncountable and $q\in\mathbb{N}$. The set of components of quasi-rank $\leq-q$ is uncountable.

1. Preliminaries and Motivation

1.1. Notations and basic results

Throughout this article let $k$ be an algebraically closed field of arbitrary characteristic. For a quiver $Q=(Q_{0},Q_{1},s,t)$, $x\in Q_{0}$ let

[TABLE]

Moreover let $n(x):=x^{+}\cup x^{-}$. If $\alpha\colon x\to y$, then by definition $s(\alpha)=x$ and $t(\alpha)=y$. We say that $\alpha$ starts in $s(\alpha)$ and ends in $t(\alpha)$.

• Definition.

A quiver $Q$ is called

1. $(a)$

locally finite if $n(x)$ is finite for each $x\in Q_{0}$, 2. $(b)$

of bounded length if for each $x\in Q_{0}$ there is $N_{x}\in\mathbb{N}$ such that each path in $Q$ which starts or ends in $x$ is of length $\leq N_{x}$, 3. $(c)$

locally bounded if $Q$ is locally finite and of bounded length.

From now on we assume that $Q$ is locally bounded. Note that this implies that $Q$ is acyclic. Moreover, we assume that $Q$ is connected. A finite dimensional representation $M=((M_{x})_{x\in Q_{0}},(M(\alpha))_{\alpha\in Q_{1}})$ over $Q$ consists of vector spaces $M_{x}$ and linear maps $M(\alpha)\colon M_{s(\alpha)}\to M_{t(\alpha)}$ such that $\dim_{k}M:=\sum_{x\in Q_{0}}\dim_{k}M_{x}$ is finite. A morphism $f\colon M\to N$ between representations is a collection of linear maps $(f_{z})_{z\in Q_{0}}$ such for each arrow $\alpha\colon x\to y$ there is a commutative diagram

[TABLE]

The category of finite dimensional representations over $Q$ is denoted by $\operatorname{rep}(Q)$. The path category $k(Q)$ has $Q_{0}$ as set of objects and $\operatorname{Hom}_{k(Q)}(x,y)$ is the vector space with basis given by the paths from $x$ to $y$. The trivial arrow in $x$ is denoted by $\epsilon_{x}$. Since $Q$ is locally bounded, the category $k(Q)$ is locally bounded in the sense of [Gab2, 2.1]. A finite-dimensional module over a locally bounded category $A$ is a functor $F\colon A\to\operatorname{mod}k$ such that $\sum_{x\in A}\dim_{k}F(x)$ is finite. The category $\operatorname{mod}A$ has Auslander-Reiten sequences $($see [Gab2, 2.2]$)$. Since a finite dimensional module over $k(Q)$ is the same as a representation of $Q$, the category $\operatorname{rep}(Q)$ has Auslander-Reiten sequences.

If moreover $Q_{0}$ is a finite set, we denote with $kQ$ the path algebra of $Q$ with idempotents $e_{x},x\in Q_{0}$. In this case $kQ$ is a finite dimensional, associative, basic and connected $k$-algebra. We denote by $\operatorname{mod}kQ$ the class of finite-dimensional $kQ$ left modules. Given $M\in\operatorname{mod}kQ$ we let $M_{x}:=e_{x}M$. The categories $\operatorname{mod}kQ$ and $\operatorname{rep}(Q)$ are equivalent $($see for example [Assem1, III 1.6]$)$. We will therefore switch freely between representations of $Q$ and modules of $kQ$, if one of the approaches seems more convenient for us. We assume that the reader is familiar with Auslander-Reiten theory and basic results on wild hereditary algebras. For a well written survey on the subjects we refer to [Assem1], [Ker2] and [Ker3]. Recall the definition of the dimension function

[TABLE]

If $0\to A\to B\to C\to 0$ is an exact sequence, then $\operatorname{\underline{dim}}A+\operatorname{\underline{dim}}C=\operatorname{\underline{dim}}B$. A finite quiver $Q$ defines a $($non-symmetric) bilinear form $\langle-,-\rangle\colon\mathbb{Z}^{Q_{0}}\times\mathbb{Z}^{Q_{0}}\to\mathbb{Z},$ given by $((x_{i}),(y_{j}))\mapsto\sum_{i\in Q_{0}}x_{i}y_{i}-\sum_{\alpha\in Q_{1}}x_{s(\alpha)}y_{t(\alpha)}$, which coincides with the Euler-Ringel form [Ri4] on the Grothendieck group $K_{0}(kQ)\cong\mathbb{Z}^{Q_{0}}$, i.e. for $M,N\in\operatorname{mod}kQ$ we have

[TABLE]

1.2. The Kronecker algebra and $\mathbb{Z}A_{\infty}$ components

We always assume that $r\geq 3$. Denote by $\Gamma_{r}$ the $r$-Kronecker quiver, which is given by two vertices $1,2$ and arrows $\gamma_{1},\ldots,\gamma_{r}\colon 1\to 2$.

We set $\mathcal{K}_{r}:=k\Gamma_{r}$ and $P_{1}:=\mathcal{K}_{r}e_{2}$, $P_{2}:=\mathcal{K}_{r}e_{1}$. The modules $P_{1}$ and $P_{2}$ are the indecomposable projective modules of $\operatorname{mod}\mathcal{K}_{r}$, $\dim_{k}\operatorname{Hom}(P_{1},P_{2})=r$ and $\dim_{k}\operatorname{Hom}(P_{2},P_{1})=0$. As Figure 1 suggests, we write $\operatorname{\underline{dim}}M=(\dim_{k}M_{1},\dim_{k}M_{2})$. For example $\operatorname{\underline{dim}}P_{1}=(0,1)$ and $\operatorname{\underline{dim}}P_{2}=(1,r)$.

Figure 2 shows the notation we use for the components $\mathcal{P},\mathcal{I}$ in the Auslander-Reiten quiver of $\mathcal{K}_{r}$ which contain the indecomposable projective modules $P_{1},P_{2}$ and indecomposable injective modules $I_{1},I_{2}$. The set of all other components is denoted by $\mathcal{R}$.

Ringel has proven [Ri3, 2.3] that every component in $\mathcal{R}$ is of type $\mathbb{Z}A_{\infty}$. A module in such a component is called regular and the class of all regular indecomposable modules is denoted by $\operatorname{ind}\mathcal{R}$. An irreducible morphism in a component of type $\mathbb{Z}A_{\infty}$ $($for any algebra$)$ is injective if the corresponding arrow is uprising and surjective otherwise. A representation $M$ in a $\mathbb{Z}A_{\infty}$ component is called quasi-simple if the AR sequence terminating in M has an indecomposable middle term. These modules are the modules in the bottom layer of the component. If $M$ is quasi-simple in a component $\mathcal{C}$ of type $\mathbb{Z}A_{\infty}$, then there is an infinite chain $($ray$)$ of irreducible monomorphisms $($resp. epimorphisms$)$

[TABLE]

[TABLE]

and for each indecomposable module $X$ in $\mathcal{C}$ there are unique quasi-simple modules $N,M$ and $l\in\mathbb{N}$ with $(l)M=X=N[l]$. The number $\operatorname{ql}(X):=l$ is called the quasi-length of $X$.

The indecomposable modules in $\mathcal{P}$ are called preprojective modules and the modules in $\mathcal{I}$ are called preinjective modules. Moreover we call an arbitrary module preprojective $($resp. preinjective, regular$)$ if all its indecomposable direct summands are preprojective $($res. preinjective, regular$)$. We have $P$ in $\mathcal{P}$ $(I$ in $\mathcal{I})$ if and only if there is $l\in\mathbb{N}_{0}$ with $\tau^{l}P=P_{i}$ $(\tau^{-l}I=I_{i})$ for $i\in\{1,2\}$. Let $\operatorname{mod}_{pf}\mathcal{K}_{r}$ be the subcategory of all modules without non-zero projective direct summands and $\operatorname{mod}_{if}\mathcal{K}_{r}$ the subcategory of all modules without non-zero injective summands. Since $\mathcal{K}_{r}$ is a hereditary algebra, the Auslander-Reiten translation $\tau\colon\operatorname{mod}\mathcal{K}_{r}\to\operatorname{mod}\mathcal{K}_{r}$ induces an equivalence from $\operatorname{mod}_{pf}\mathcal{K}_{r}$ to $\operatorname{mod}_{if}\mathcal{K}_{r}$, that will be often used later on without further notice.

1.3. The connection between $\operatorname{rk}(\mathcal{C})$ and $\mathcal{W}(\mathcal{C})$

Let us start by recalling the definitions of $\operatorname{rk}(\mathcal{C})$ and $\mathcal{W}(\mathcal{C})$.

• Definition.

Let $A=kQ$ be a wild hereditary algebra and $\mathcal{C}$ be a regular component with a quasi-simple module $X$ in $\mathcal{C}$, then

[TABLE]

It was shown in [Ker1], that $\overline{t}(A):=\max\{\operatorname{rk}(\mathcal{C})\mid\mathcal{C}\ \text{regular component}\}$ is finite and $\overline{t}(\mathcal{K}_{r})=1$. In other words, there are lots of morphisms in $\tau$-direction. Also in $\tau^{-1}$-direction one finds a lot of morphisms since $\underline{t}(A):=\inf\{\operatorname{rk}(\mathcal{C})\mid\mathcal{C}\ \text{regular component}\}=-\infty$ $($see [KerLuk2, 3.1]$)$.

Another invariant that can be attached to a regular component $\mathcal{C}$ of the Kronecker algebra is defined in [Wor1] and denoted by $\mathcal{W}(\mathcal{C})$. In order to define $\mathcal{W}(\mathcal{C})$, we first recall the definitions of $\operatorname{EKP}$ and $\operatorname{EIP}$. For $\alpha\in k^{r}\setminus\{0\}$ and $M\in\operatorname{rep}(\Gamma_{r})$ we consider the $k$-linear map $M^{\alpha}:=\sum^{r}_{i=1}\alpha_{i}M(\gamma_{i})\colon M_{1}\to M_{2}$ and let $X_{\alpha}\in\operatorname{rep}(\Gamma)$ be the cokernel of the embedding $(0,\iota_{2})\colon P_{1}\to P_{2}$ where $\iota_{2}\colon k\to k^{r}$, $x\mapsto x\alpha$.

• Definition.

[Wor1, 2.1] We define the classes of representations with the equal kernels property and with the equal images property as

1. (1)

$\operatorname{EKP}:=\{M\in\operatorname{rep}(\Gamma_{r})\mid\forall\alpha\in k^{r}:M^{\alpha}\ \text{is injective}\}$ and 2. (2)

$\operatorname{EIP}:=\{M\in\operatorname{rep}(\Gamma_{r})\mid\forall\alpha\in k^{r}:M^{\alpha}\ \text{is surjective}\}.$

Given a regular component $\mathcal{C}$, there exist uniquely determined quasi-simple representations $M_{\mathcal{C}}$ and $W_{\mathcal{C}}$ in $\mathcal{C}$ such that $(M_{\mathcal{C}}\to)=\operatorname{EKP}\cap\mathcal{C}$ and $(\to W_{\mathcal{C}})=\operatorname{EIP}\cap\mathcal{C}$ $($see [Wor1, 3.3]$)$. Now $\mathcal{W}(\mathcal{C})$ is defined as the unique integer with $\tau^{\mathcal{W}(\mathcal{C})+1}M_{\mathcal{C}}=W_{\mathcal{C}}$. Since $\operatorname{EKP}\cap\operatorname{EIP}=\{0\}$ it follows $\mathcal{W}(\mathcal{C})\geq 0$.

Proposition 1.3.1**.**

Let $\mathcal{C}$ be a regular component, then $-\mathcal{W}(\mathcal{C})\leq\operatorname{rk}(\mathcal{C})\leq-\mathcal{W}(\mathcal{C})+3$.

Proof.

On the one hand let $M:=\tau^{-1}W_{\mathcal{C}}$, then there is $\alpha\in k^{r}\setminus\{0\}$ with $0\neq\operatorname{Ext}(X_{\alpha},M)$ [Wor1, 2.5]. The Auslander-Reiten formula [Ker3, 2.3] yields $0\neq\operatorname{Hom}(\tau^{-1}M,X_{\alpha})$. On the other hand let $N:=\tau M_{\mathcal{C}}$, then there is $\beta\in k^{r}\setminus\{0\}$ with $0\neq\operatorname{Hom}(X_{\beta},N)\cong\operatorname{Hom}(\tau X_{\beta},\tau N)$ [Wor1, 2.5]. By the Euler-Ringel form we have

[TABLE]

Hence $0\neq\dim_{k}\operatorname{Ext}(X_{\beta},X_{\alpha})=\dim_{k}\operatorname{Hom}(\tau^{-1}X_{\alpha},X_{\beta})=\dim_{k}\operatorname{Hom}(X_{\alpha},\tau X_{\beta})$. Since $X_{\alpha}$ and $\tau X_{\beta}$ are elementary [Bi1, 2.1.4], we get a non-zero morphism by [Bi1, 2.1.1]

[TABLE]

[TABLE]

Hence $\operatorname{rad}(W_{\mathcal{C}},\tau^{-\mathcal{W}(\mathcal{C})+3}W_{\mathcal{C}})\neq 0$, since [Wor1, 4.10] together with $\mathcal{W}(\mathcal{C})=3$ implies that $W_{\mathcal{C}}$ is not a brick. By [Ker1, 1.7] it follows that $\operatorname{rk}(\mathcal{C})\leq-\mathcal{W}(\mathcal{C})+3$. The second inequality follows from the proof of [Wor1, 3.1.3]. ∎

In [Wor1], the inequality $-\mathcal{W}(\mathcal{C})\leq\operatorname{rk}(\mathcal{C})$ in conjunction with $\underline{t}(\mathcal{K}_{r})=-\infty$ was used to prove that $\sup\{\mathcal{W}(\mathcal{D})\mid\mathcal{D}\in\mathcal{R}\}=\infty$. We choose a different approach and study the number $\mathcal{W}(\mathcal{C})$ to draw conclusions for $\operatorname{rk}(\mathcal{C})$.

• Remark.

There are regular components with $\operatorname{rk}(\mathcal{C}_{i})=1$ and $\mathcal{W}(\mathcal{C}_{i})=i$ for $0\leq i\leq 2$ [Wor1, 3.3.1] and a component $\mathcal{D}$ with $\operatorname{rk}(\mathcal{D})=0$ and $\mathcal{W}(\mathcal{D})=0$ [Bi1, 3.3.3].

2. Covering Theory

2.1. General Theory

We follow [Ri6] and [Ri7] and consider the universal cover $C_{r}$ of the quiver $\Gamma_{r}$. The underlying graph of $C_{r}$ is an $r$-regular tree and $C_{r}$ has bipartite orientation. That means each vertex $x\in(C_{r})_{0}$ is a sink or a source and $|n(x)|=r$. In the following we recall the construction of $C_{r}$.

For a quiver $Q=(Q_{0},Q_{1},s,t)$ with arrow set $Q_{1}$ we write $(Q_{1})^{-1}:=\{\alpha^{-1}\mid\alpha\in Q_{1}\}$ for the formal inverses of $Q_{1}$. Moreover we extend the functions $s$ and $t$ to $(Q_{1})^{-1}$ by defining $s(\alpha^{-1}):=t(\alpha)$ and $t(\alpha^{-1}):=s(\alpha)$. A walk $w$ in $Q_{1}$ is a formal sequence $w=\alpha^{\varepsilon_{n}}_{n}\cdots\alpha^{\varepsilon_{1}}_{1}$ with $\alpha_{i}\in Q_{1}$, $\varepsilon\in\{1,-1\}$ such that $s(\alpha^{\varepsilon_{i+1}}_{i+1})=t(\alpha^{\varepsilon_{i}}_{i})$ for all $i<n$, where $\alpha^{1}:=\alpha$ for all $\alpha\in Q_{1}$. We set $t(w):=t(\alpha^{\varepsilon_{n}}_{n})$ and $s(w):=s(\alpha^{\varepsilon_{1}}_{1})$.

Let $\sim$ be the equivalence relation on the set of walks $W$ of $\Gamma_{r}$ generated by

[TABLE]

Let ${}^{-1}\colon W\to W$ be the involution on $W$ given by $(\alpha^{\varepsilon_{n}}_{n}\cdots\alpha^{\varepsilon_{1}}_{1})^{-1}:=\alpha^{-\varepsilon_{1}}_{1}\cdots\alpha^{-\varepsilon_{n}}_{n}$. Now consider the fundamental group $\pi(\Gamma_{r})$ of $\Gamma_{r}$ in the point $1$, i.e. the elements of $\pi(\Gamma_{r})$ are the equivalence classes of unoriented paths starting and ending in $1$, with multiplication given by concatenation of paths, inverse elements $[w]^{-1}:=[w^{-1}]$ and identity element $[\epsilon_{1}]$. Note that $\pi(\Gamma_{r})$ is a free group in the $r-1$ generators $\{[\gamma^{-1}_{j}\gamma_{1}]\mid 2\leq j\leq r\}$ and in particular torsionfree.

The quiver $C_{r}$ is given by the following data:

1. $(a)$

$(C_{r})_{0}$ is the set of equivalence classes of paths starting in $1$. 2. $(b)$

There is an arrow from $[w]$ to $[w^{\prime}]$ whenever $w^{\prime}\sim\gamma_{i}w$ for some $i\in\{1,\ldots,r\}$.

Let $\pi\colon C_{r}\to\Gamma_{r}$ be the quiver morphism given by $[w]\mapsto t(w)$ and $([w]\to[\gamma_{i}w])\mapsto\gamma_{i}$. The morphisms $\pi$ is a $G$-Galois cover for $G=\pi(\Gamma_{r})$, where the action of $G$ on $C_{r}$ is given by concatenation of paths: if $g=[w]\in\pi(\Gamma_{r})$ and $[v],[u]\in(C_{r})_{0}$ with arrow $[u]\to[\gamma_{i}u]$ then

[TABLE]

[TABLE]

We define $C_{r}^{+}:=\pi^{-1}(\{1\})$, $C_{r}^{-}:=\pi^{-1}(\{2\})$ and get an induced action on $\operatorname{rep}(C_{r})$ by shifting the support of representations via $G=\pi(\Gamma_{r})$: Given $M\in\operatorname{rep}(C_{r})$ and $g\in G$ we define $M^{g}:=(((M^{g})_{x})_{x\in(C_{r})_{0}},(M^{g}(\alpha))_{\alpha\in(C_{r})_{1}})$, where

[TABLE]

By identifying the orbit quiver $C_{r}/G$ with $\Gamma_{r}$ we define the push-down functor $\pi_{\lambda}\colon\operatorname{rep}(C_{r})\to\operatorname{rep}(\Gamma_{r})$ on the objects via $\pi_{\lambda}(M):=(\pi_{\lambda}(M)_{1},\pi_{\lambda}(M)_{2};(\pi_{\lambda}(M)(\gamma_{i}))_{1\leq i\leq r})$, where

[TABLE]

[TABLE]

If $f=(f_{x})_{x\in(C_{r})_{0}}\colon M\to N$ is a morphism in $\operatorname{rep}(C_{r})$ then $\pi_{\lambda}(f)=(g_{\pi(x)})_{x\in(C_{r})_{0}}=(g_{1},g_{2})$ with

[TABLE]

By [Gab2, 3.2] $\pi_{\lambda}$ is an exact functor.

Theorem 2.1.1**.**

[Gab3, 3.6]**, [Ri7, 6.2,6.3] The following statements hold.

1. $(a)$

$\pi_{\lambda}$* sends indecomposable representations in $\operatorname{rep}(C_{r})$ to indecomposable representations in $\operatorname{rep}(\Gamma_{r})$.* 2. $(b)$

If $M\in\operatorname{rep}(C_{r})$ is indecomposable then $\pi_{\lambda}(M)\cong\pi_{\lambda}(N)$ if and only if $M^{g}\cong N$ for some $g\in G$. 3. $(c)$

$\pi_{\lambda}$* sends $AR$ sequences to $AR$ sequences and $\pi_{\lambda}$ commutes with the Auslander-Reiten translates, i.e. $\tau\circ\pi_{\lambda}=\pi_{\lambda}\circ\tau_{C_{r}}$.* 4. $(d)$

If $M\in\operatorname{rep}(C_{r})$ is indecomposable in a component $\mathcal{D}$ with $\pi_{\lambda}(M)$ in a component $\mathcal{C}$, then $\pi_{\lambda}$ induces a covering $\mathcal{D}\to\mathcal{C}$ of translation quivers.

• Definition.

Let $M\in\operatorname{rep}(C_{r})$ be indecomposable. $M$ is called regular if $\pi_{\lambda}(M)$ is regular. A component $\mathcal{D}$ of $\operatorname{rep}(C_{r})$ is called regular if it contains a regular representation $M$. In this case we denote by $\pi_{\lambda}(\mathcal{D})$ the component containing $\pi_{\lambda}(M)$. Moreover we let $\mathcal{R}({C_{r}})$ be the set of all regular components of the Auslander-Reiten quiver of $\operatorname{rep}(C_{r})$.

Corollary 2.1.2**.**

Let $\mathcal{D}$ be regular component, then the covering $\mathcal{D}\to\pi_{\lambda}(\mathcal{D})$ is an isomorphism of translation quivers. In particular, $\mathcal{D}$ is of type $\mathbb{Z}A_{\infty}$. Moreover a component $\mathcal{E}$ is regular if and only if $\mathcal{E}$ is of type $\mathbb{Z}A_{\infty}$.

Proof.

By [Gab2, 1.7] $\pi_{\lambda}(\mathcal{D})\cong\mathbb{Z}A_{\infty}$ is a simply connected translation quiver. By [Gab2, 1.6],[Ried1, 1.7] the quiver morphism $\mathcal{D}\to\pi_{\lambda}(\mathcal{D})$ is an isomorphism. If $\pi_{\lambda}(\mathcal{E})\in\{\mathcal{I},\mathcal{P}\}$ then there exists a vertex $x$ with $r\geq 3$ successors $($see Figure $\ref{Fig:AuslanderReitenquiver})$, since a covering is surjective on arrows. Hence $\mathcal{E}$ is not of type $\mathbb{Z}A_{\infty}$. ∎

2.2. Duality

Recall [Wor1, 2.2] that the duality $D\colon\operatorname{rep}(\Gamma_{r})\to\operatorname{rep}(\Gamma_{r})$ is defined by setting $(DM)_{x}:=(M_{\psi(x)})^{\ast}$ and $(DM)(\gamma_{i}):=(M(\gamma_{i}))^{\ast}$, where $\psi\colon\{1,2\}\to\{1,2\}$ is the involution with $\psi(1)=2$.

We define an involution $\varphi_{0}\colon(C_{r})_{0}\to(C_{r})_{0}$ via $[w]\mapsto[\overline{w}\gamma_{1}]$, where $\overline{\epsilon_{1}}:=\epsilon_{2}$, $\overline{\epsilon_{2}}:=\epsilon_{1}$ and $\overline{\alpha^{\varepsilon_{n}}_{n}\cdots\alpha^{\varepsilon_{1}}_{1}}:=\alpha^{-\varepsilon_{n}}_{n}\cdots\alpha^{-\varepsilon_{1}}_{1}$. This induces a quiver anti-morphism $\varphi\colon C_{r}\to C_{r}$ in the following way.

If $[w]\to[\gamma_{i}w]$ is an arrow of $C_{r}$, then by definition there is a unique arrow $\varphi([w]\to[\gamma_{i}w])$ starting in $\varphi_{0}([\gamma_{i}w])=[\gamma^{-1}_{i}\overline{w}\gamma_{1}]$ and ending in $\varphi_{0}([w])=[\overline{w}\gamma_{1}]$, since $\gamma_{i}\gamma^{-1}_{i}\overline{w}\gamma_{1}\sim\overline{w}\gamma_{1}$. Note that $\varphi(C_{r}^{+})$ = $C_{r}^{-}$, $\varphi(C_{r}^{-})=C_{r}^{+}$ and $\pi(\varphi(\alpha))=\pi(\alpha)$.

We define a duality $D_{C_{r}}\colon\operatorname{rep}(C_{r})\to\operatorname{rep}(C_{r})$ by setting $D_{C_{r}}M:=((D_{C_{r}}M)_{x\in(C_{r})_{0}},(D_{C_{r}}M(\alpha))_{\alpha\in Q_{1}})$ where $(D_{C_{r}}M)_{x}:=(M_{\varphi(x)})^{\ast}$ and $D_{C_{r}}M(\alpha):=(M(\varphi(\alpha)))^{\ast}$. By construction we have $\pi_{\lambda}\circ D_{C_{r}}=D\circ\pi_{\lambda}$.

3. Lifting EKP and EIP to $\operatorname{rep}(C_{r})$

In the following we give a characterization of the equal images and equal kernels property for indecomposable representations of the form $\pi_{\lambda}(M)$. Let $\bar{}\ \colon(C_{r})_{1}\to\{1,\ldots,r\}$ be the unique map with $\gamma_{\overline{\beta}}=\pi(\beta)$ for all $\beta\in(C_{r})_{1}$. Note that if $x\in(C_{r})^{+}($or $x\in C_{r}^{-})$ then the restriction of \ \bar{}\ to $\{\alpha\in(C_{r})_{1}\mid s(\alpha)=x\}$ $($resp. $\{\alpha\in(C_{r})_{1}\mid t(\alpha)=x\})$ is a bijective map to $\{1,\ldots,r\}$.

• Definition.

Let $\varnothing\neq X\subseteq(C_{r})_{0}$ be a set of vertices and $T\subseteq C_{r}$ be a tree.

1. $(a)$

The unique minimal tree containing $X$ is denoted by $T(X)$. 2. $(b)$

A vertex $x\in T_{0}$ is called a leaf of $T$, if $|n(x)\cap T_{0}|\leq 1$.

• Definition.

Let $x\in(C_{r})_{0}$ and $M$ be a representation of $C_{r}$.

1. $(a)$

The set $\operatorname{supp}(M):=\{y\in(C_{r})_{0}\mid M_{y}\neq 0\}$ is called the support of $M$. 2. $(b)$

For $V\subseteq\operatorname{supp}(M)$ we let $M_{V}$ be the induced representation with $\operatorname{supp}(M_{V})=V$. 3. $(c)$

The vertex $x$ is a leaf of $M$ if $x$ is a leaf of $T(M):=T(\operatorname{supp}(M))$. 4. $(d)$

$M$ is called balanced provided that $M$ is indecomposable and $M$ has leaves in $C_{r}^{+}$ and $C_{r}^{-}$.

Observe that if $M$ is indecomposable we have $T(M)_{0}=\operatorname{supp}(M)$.

• Definition.

We define

[TABLE]

[TABLE]

Theorem 3.1**.**

Let $M\in\operatorname{rep}(C_{r})$ be an indecomposable representation. The following statements are equivalent:

1. $(a)$

$N:=\pi_{\lambda}(M)$* has the equal kernels property.* 2. $(b)$

$N(\gamma_{i})$* is injective for all $i\in\{1,\ldots,r\}$.* 3. $(c)$

$M\in\operatorname{Inj}$.

Proof.

$(a)\Rightarrow(b)$: Clear from the definition of $\operatorname{EKP}$.

$(b)\Rightarrow(c)$: Let $\alpha\colon x\to y$ and $m_{x}\in\ker M(\alpha)$. Denote by $\iota_{x}\colon M_{x}\to\bigoplus_{z\in C_{r}^{+}}M_{z}$ the natural embedding and let $\iota_{y}\colon M_{y}\to\bigoplus_{z\in C_{r}^{-}}M_{z}$. We conclude

[TABLE]

Hence $\iota_{x}(m_{x})\in\ker N(\gamma_{\overline{\alpha}})=\{0\}$ and $m_{x}=0$.

$(c)\Rightarrow(a)$: Let $\alpha\in k^{r}\setminus\{0\}$, $f:=\sum^{r}_{i=1}\alpha_{i}N(\gamma_{i})\colon N_{1}\to N_{2}$ and $m\in\ker f$. We assume w.l.o.g that $\alpha_{1}\neq 0$. Write $m=(m_{z})_{z\in C_{r}^{+}}$. We have to show that $m_{z}=0$ for all $z\in C_{r}^{+}$.

Let $S:=\{z\in(C_{r})_{0}\mid m_{z}\neq 0\}\subseteq C_{r}^{+}$ and suppose that $S\neq\varnothing$. Let $T(S)$ be the minimal tree that contains $S$. Since $T(S)_{0}\subseteq\operatorname{supp}(M)$ every leaf belongs to $C^{+}_{r}\cap S$, by the minimality of $T(S)$. Fix a leaf $x$ in $T(S)$ and let $\gamma\colon x\to y$ be the unique arrow with $\pi(\gamma)=\gamma_{1}$. Then

[TABLE]

By the injectivity of $M(\gamma)$ there is $\delta\colon z\to y\in(C_{r})_{1}\setminus\{\gamma\}$ with $t(\delta)=y$ and $0\neq\alpha_{\overline{\delta}}M(\delta)(m_{z})$. It follows $m_{z}\neq 0$ and $z,x\in S$. Since $C_{r}$ is a tree we get $y\in T(S)_{0}$.

Since $\overline{\delta}\neq\overline{\gamma}=1$, we assume without loss of generality that $\overline{\delta}=2$, so that $\alpha_{2}\neq 0$. Let $\eta\colon x\to a$ be the unique arrow with $\overline{\eta}=2$. Then

[TABLE]

Hence there is $\zeta\colon b\to a\in(C_{r})_{1}\setminus\{\eta\}$ with $0\neq\alpha_{\overline{\zeta}}M(\zeta)(m_{b})$ and $a,b$ are in $T(S)_{0}$. We have shown that $a$ and $y$ are in $T(S)_{0}$. This is a contradiction since $x$ is a leaf. ∎

Corollary 3.2**.**

Let $M\in\operatorname{rep}(C_{r})$ be an indecomposable representation. The following statements are equivalent.

1. $(a)$

$N:=\pi_{\lambda}(M)$* has the equal images property.* 2. $(b)$

$N(\gamma_{i})$* is surjective for all $i\in\{1,\ldots,r\}$.* 3. $(c)$

$M\in\operatorname{Surj}$.

Proof.

$(c)\Rightarrow(a)$: Let $M(\alpha)$ be surjective for each $\alpha\in(C_{r})_{1}$, then $(D_{C_{r}}M)(\alpha)$ is injective for each $\alpha\in(C_{r})_{1}$. By 3.1, the representation $\pi_{\lambda}(D_{C_{r}}M)\cong D\pi_{\lambda}(M)$ has the equal kernels property. Therefore $\pi_{\lambda}(M)$ has the equal images property, since $D(\operatorname{EKP})=\operatorname{EIP}$. ∎

Corollary 3.3**.**

Let $\mathcal{D}\in\mathcal{R}(C_{r})$ and $\mathcal{C}:=\pi_{\lambda}(\mathcal{D})$. Then there exist uniquely determined quasi-simple representations $I_{\mathcal{D}}$ and $S_{\mathcal{D}}$ in $\mathcal{D}$ such that

1. $(a)$

$\pi_{\lambda}(I_{\mathcal{D}})=M_{\mathcal{C}}$* and $\pi_{\lambda}(S_{\mathcal{D}})=W_{\mathcal{C}}$.* 2. $(b)$

$\operatorname{Surj}\cap\mathcal{D}=(\to S_{\mathcal{D}})$* and $\operatorname{Inj}\cap\mathcal{D}=(I_{\mathcal{D}}\to)$.* 3. $(c)$

The unique integer $\mathcal{W}_{C}(\mathcal{D})$ with $\tau_{C_{r}}^{\mathcal{W}_{C}(\mathcal{D})+1}(I_{\mathcal{D}})=S_{\mathcal{D}}$ is given by $\mathcal{W}_{C}(\mathcal{D})=\mathcal{W}(\mathcal{C})\in\mathbb{N}_{0}$.

Corollary 3.4**.**

Assume that $M\in\operatorname{rep}(C_{r})$ is balanced, then $M$ is regular.

Proof.

If an indecomposable representation $X\in\operatorname{rep}(C_{r})$ is in $\operatorname{Inj}$ $($respectively $\operatorname{Surj})$, then all leaves of $X$ are in $C_{r}^{+}$ $($resp. $C_{r}^{-})$. Hence $\pi_{\lambda}(M)\notin\operatorname{EIP}\cup\operatorname{EKP}$ by 3.1. By [Wor1, 2.7] the representations of the components $\mathcal{P}$ and $\mathcal{I}$ are contained in $\operatorname{EKP}\cup\operatorname{EIP}$.∎

From now on we write $x_{0}:=[\epsilon_{1}]\in(C_{r})_{0}$ for the vertex in $(C_{r})_{0}$ given by the trivial walk starting in the vertex $1$.

• Definition.

For $i\in\{1,\ldots,r\}$, we let $\beta_{i}\in(C_{r})_{1}$ be the unique arrow with $s(\beta_{i})=x_{0}$ and $\pi(\beta_{i})=\gamma_{i}$. Moreover let $z_{i}:=t(\beta_{i})$. We define an indecomposable representation $X^{i}$ in $\operatorname{rep}(C_{r})$ via:

[TABLE]

and $X^{i}(\beta_{j}):=\operatorname{id}_{k}$ for all $j\neq i$. By definition we have $\operatorname{\underline{dim}}\pi_{\lambda}(X^{i})=(1,r-1)$ and $\pi_{\lambda}(X^{i})\cong X_{e_{i}}$.

In view of [Wor1, 2.5], [Gab3, 3.6(c)] 2.1.1 and 3.1 we conclude the following.

Corollary 3.5**.**

Let $M\in\operatorname{rep}(C_{r})$ be an indecomposable representation. The following statements are equivalent:

1. $(a)$

$N:=\pi_{\lambda}(M)$* has the equal kernels property.* 2. $(b)$

$\operatorname{Hom}(\pi_{\lambda}(X^{i}),N)=0$* for all $i\in\{1,\ldots,r\}$.* 3. $(c)$

$\operatorname{Hom}({(X^{i})}^{g},M)=0$* for all $i\in\{1,\ldots,r\}$ and all $g\in G$.*

The following Lemma will be needed later on.

Lemma 3.6**.**

Let $n\in\mathbb{N}$, $M$ be regular indecomposable, $i\in\{1,\ldots,r\}$ and $g\in G=\pi(\Gamma_{r})$.

1. $(a)$

For $n\geq 2$, the linear map $(\tau^{n}_{C_{r}}X^{i})^{g}(\alpha)$ is surjective for all $\alpha\in(C_{r})_{1}$. 2. $(b)$

If $n\geq 1$ and $f=(f_{x})_{x\in(C_{r})_{0}}\colon(\tau^{n}_{C_{r}}X^{i})^{g}\to M$ is a non-zero morphism, then each $f_{x}$ is injective. 3. $(c)$

If $n\geq 2$, $x\in\operatorname{supp}((\tau^{n}_{C_{r}}X^{i})^{g})\cap C_{r}^{-}$ and $0\neq f\colon(\tau^{n}_{C_{r}}X^{i})^{g}\to M$, then $\operatorname{supp}(M)\cap n(x)=\operatorname{supp}(M)\cap x^{-}=x^{-}$. This means $|\operatorname{supp}(M)\cap x^{-}|=r$.

Proof.

$(a)$ For $n\geq 2$, we have $\pi_{\lambda}((\tau^{n}_{C_{r}}X^{i})^{g})=\pi_{\lambda}(\tau^{n}_{C_{r}}X^{i})=\tau^{n}X_{e_{i}}\in\operatorname{EIP}$ $($see [Wor1, 3.3]$)$.

$(b)$ For $n\geq 1$, it is known that each proper factor of $\tau^{n}X_{e_{i}}$ is preinjective, see for example [Bi1, 2.1.4]. Let $0\neq f\in\operatorname{Hom}((\tau_{C_{r}}^{n}X^{i})^{g},M)$, then $0\neq\pi_{\lambda}(f)=(g_{i})_{1\leq i\leq 2}\colon\tau^{n}X_{e_{1}}\to\pi_{\lambda}(M)$ is injective, where

[TABLE]

So each $f_{x}$ is injective, since $\pi^{-1}(\{1,2\})=(C_{r})_{0}$.

$(c)$ Let $x\in\operatorname{supp}((\tau^{n}_{C_{r}}X^{i})^{g})\cap C_{r}^{-}$ and $z\in n(x)$. Since $x$ is a sink, there is $\alpha\colon z\to x$ and by $(a)$, $(\tau^{n}_{C_{r}}X^{i})(\alpha)$ is surjective. Hence $((\tau^{n}_{C_{r}}X^{i})^{g})_{z}\neq 0$. By $(b)$, we get $M_{z}\neq 0$ and $z\in\operatorname{supp}(M)\cap x^{-}$. ∎

4. Considerations in the universal covering

Let us recall what we have shown so far. Given a component $\mathcal{D}$ in $\mathcal{R}(C_{r})$, the natural number $\mathcal{W}_{C}(\mathcal{D})$ is the distance between the two non-empty, non-intersecting cones $\operatorname{Inj}\cap\mathcal{D}$ and $\operatorname{Surj}\cap\mathcal{D}$. Moreover we know that $\mathcal{W}(\pi_{\lambda}(\mathcal{D}))=\mathcal{W}_{C}(\mathcal{D})$.

Let now $X\in\mathcal{D}$ be indecomposable. Then there exists an integer $l\in\mathbb{Z}$ such that $\tau^{l}_{C_{r}}X\in\operatorname{Surj}$, since $\operatorname{Surj}\cap\mathcal{D}$ is non-empty. We also find $n\geq 1$ with $\tau^{-n}_{C_{r}}X\not\in\operatorname{Surj}$. Since $\operatorname{Surj}\cap\mathcal{D}$ is closed under $\tau_{C_{r}}$ we conclude $-n\leq l$. Therefore the following minima exist.

• Definition.

Let $X\in\operatorname{rep}(C_{r})$ be a regular indecomposable representation. We define

[TABLE]

Note that $|d^{-}(X)|\in\mathbb{N}_{0}$ is the distance of $X$ to the border of the cone $\operatorname{Inj}\cap\mathcal{D}$.

Lemma 4.1**.**

Let $X\in\operatorname{rep}(C_{r})$ be indecomposable in a regular component $\mathcal{D}$. Then

[TABLE]

Proof.

Since the equality is obvious for $X$ quasi-simple we assume $l:=\operatorname{ql}(X)>1$. Let $Z$ be the unique quasi-simple representation with $X=Z[l]$. By induction we get $\mathcal{W}_{C}(\mathcal{D})=d^{+}(Z[l-1])+d^{-}(Z[l-1])-(l-1)$. Now observe that $d^{+}(Z[l-1])+1=d^{+}(Z[l])$ and $d^{-}(Z[l-1])=d^{-}(Z[l])$. Hence $\mathcal{W}_{C}(\mathcal{D})=d^{+}(Z[l])-1+d^{-}(Z[l])-(l-1)=d^{+}(X)+d^{-}(X)-\operatorname{ql}(X)$. ∎

We show in the following how to modify a regular representation $X\rightsquigarrow X^{\prime}$ such that the obtained representation $X^{\prime}$ is regular, $d^{+}(X)=d^{+}(X^{\prime})$, $d^{-}(X)=d^{-}(X^{\prime})$ and $\operatorname{ql}(X^{\prime})=1$.

4.1. Indecomposable representations arising from extensions

In this section, we show how to construct non-split exact sequences with indecomposable middle term in $\operatorname{rep}(C_{r})$.

• Definition.

Let $M,N\in\operatorname{rep}(C_{r})$ be indecomposable. The pair $(N,M)$ is called leaf-connected if there is $\alpha\colon x\to y\in(C_{r})_{1}$ s.t.

1. $(a)$

$x$ is a leaf of $M$, $y$ is a leaf of $N$ and 2. $(b)$

$\operatorname{supp}(M)\cap\operatorname{supp}(N)=\varnothing$.

• Remark.

Note that the assumption $M$ and $N$ being indecomposable together with properties $(a)$ and $(b)$ already implies the uniqueness of $\alpha$. If $(N,M)$ is leaf-connected, $\alpha$ is called the connecting arrow and $(M,N)$ is not leaf-connected.

• Definition.

Let $(N,M)$ be leaf-connected with connecting arrow $\alpha\colon x\to y$ and $f\colon M_{x}\to N_{y}$ a non-zero linear map. We define a representation $N\ast_{f}M\in\operatorname{rep}(C_{r})$ by setting

[TABLE]

with $(N\ast_{f}M)_{\operatorname{supp}(X)}=X$ for $X\in\{M,N\}$ and $(N\ast_{f}M)(\alpha):=f\colon M_{x}\to N_{y}$. Moreover we denote by $\iota_{f}\colon N\to N\ast_{f}M$ and $\pi_{f}\colon N\ast_{f}M\to M$ the natural morphisms of quiver representations. The $k$-linear map along the connecting arrow is called a connecting map for $(N,M)$.

• Remark.

Note that $N\ast_{f}M$ is just an extension of $M$ by $N$ with corresponding exact sequence $\delta_{f}\colon 0\to N\stackrel{{\scriptstyle\iota_{f}}}{{\to}}N\ast_{f}M\stackrel{{\scriptstyle\pi_{f}}}{{\to}}M\to 0$. The next lemma shows that $M\ast_{f}N$ is indecomposable. In particular, $\delta_{f}$ does not split. One can also show that the map

[TABLE]

is an isomorphism of vector spaces, where $[\delta_{0}]$ is the neutral element in the abelian group $\operatorname{Ext}(M,N)$ with respect to the Baer sum.

Lemma 4.1.1**.**

Let $(N,M)$ be leaf-connected.

1. $(a)$

The representation $N\ast_{f}M$ is indecomposable. 2. $(b)$

If $M$ and $N$ are regular, then $N\ast_{f}M$ is regular.

Proof.

$(a)$ Let $U_{1},U_{2}\in\operatorname{rep}(C_{r})$ with $N\ast_{f}M=U_{1}\oplus U_{2}$. Hence we get

[TABLE]

Since $M\in\operatorname{rep}(C_{r})$ is indecomposable, there is a unique $i\in\{1,2\}$ with $(U_{i})_{\operatorname{supp}(M)}=M$. We assume $i=1$. By the same token there is a unique $j$ with $(U_{j})_{\operatorname{supp}(N)}=N$. Since $(M\ast_{f}N)(\alpha)=f\neq 0$, it follows that $j=i=1$. Hence $U_{2}=(0)$ and $N\ast_{f}M$ is indecomposable.

$(b)$ By construction $N$ is a subrepresentation of $M\ast_{f}N$ and $M$ a factor representation. Since $\pi_{\lambda}(N\ast_{f}M)$ is indecomposable with regular factor representation $\pi_{\lambda}(M)$, it is regular itself or preprojective. By the same token $\pi_{\lambda}(N\ast_{f}M)$ is regular or preinjective. ∎

Lemma 4.1.2**.**

Let $M,N\in\operatorname{rep}(C_{r})$ be indecomposable and assume $x\in C_{r}^{+}$ is a leaf of $M$ and $y\in C_{r}^{-}$ is a leaf of $N$. Then there exists $g\in G$ such that $(N^{g},M)$ is leaf-connected with connecting arrow $x\to g.y$.

Proof.

Since $x,y$ are leaves, we find at most one arrow $\delta_{M}\colon x\to x_{1}$ and at most one arrow $\delta_{N}\colon y_{1}\to y$ with $x_{1}\in\operatorname{supp}(M)$ and $y_{1}\in\operatorname{supp}(N)$. Since $r\geq 3$ we find an arrow $\alpha\colon x\to z$ with $\pi(\delta_{M})\neq\pi(\alpha)\neq\pi(\delta_{N})$. In particular $\alpha\neq\delta_{M}$ and $x_{1}\neq z$.

Now let $g\in\pi(\Gamma_{r})$ be the unique element with $g.z=y$. Then $z\in n(x)$ is a leaf of $N^{g}$. By construction, we have $\pi(g.\alpha)=\pi(\alpha)\neq\pi(\delta_{N})$ and conclude $x\notin\operatorname{supp}(N^{g})$. It follows that $\operatorname{supp}(M)\cap\operatorname{supp}(N^{g})=\varnothing$. Hence $(N^{g},M)$ is leaf-connected with connecting arrow $\alpha$. ∎

• Definition.

Let $n\geq 2$ and $M_{1},\ldots,M_{n}\in\operatorname{rep}(C_{r})$ be indecomposable. The tuple $(M_{n},\ldots,M_{1})$ is called leaf-connected, provided that $(M_{i+1},M_{i})$ is leaf-connected for all $1\leq i<n$. A tuple $(f_{n-1},\ldots,f_{1})$ of $k$-linear maps is called a connecting map for $(M_{n},\ldots,M_{1})$ if $f_{i}$ is a connecting map for $(M_{i+1},M_{i})$ for all $1\leq i<n$.

• Remark.

Let $(M,L)$ and $(L,N)$ be leaf-connected with connecting maps $f$ and $g$, then $\operatorname{supp}(M)\cap\operatorname{supp}(N)=\varnothing$, since $C_{r}$ is a tree. Hence $\operatorname{supp}(M\ast_{f}L)\cap\operatorname{supp}(N)=\varnothing$ and $(M\ast_{f}L,N)$ is connected. Therefore the next definition is well-defined.

• Definition.

Let $n\geq 2$, $(M_{n},\ldots,M_{1})$ be leaf-connected with connecting map $(f_{n-1},\ldots,f_{1})$. Then we define inductively $M_{n}\ast_{f_{n-1}}M_{n-1}\ast_{f_{n-2}}\cdots\ast_{f_{i}}M_{i}:=(M_{n}\ast_{f_{n-1}}M_{n-1}\ast\cdots\ast_{f_{i+1}}M_{i+1})\ast_{f_{i}}M_{i}$ for all $2\leq i<n$.

From now on, we assume that $(M_{n},\ldots,M_{1})$ is leaf-connected with connecting map $(f_{n-1},\ldots,f_{1})$. For $1\leq i\leq n$, we define $\ast_{j\geq i}M_{j}:=M_{n}\ast_{f_{n-1}}M_{n-1}\ast\cdots\ast_{f_{i}}M_{i}$ and $\ast_{j\leq i}M_{j}:=M_{i}\ast_{f_{i-1}}M_{i-1}\ast\cdots\ast_{f_{1}}M_{1}$. Moreover, we set $\ast_{j\geq n}M_{j}=M_{n}$ and $\ast_{j\leq 1}M_{j}=M_{1}$.

Lemma 4.1.3**.**

Let $n\geq 2$, $(M_{n},\ldots,M_{1})$ be leaf-connected and $1\leq i<n$.

1. $(a)$

The representation $\ast_{j\geq i}M_{j}$ is indecomposable. 2. $(b)$

There is a short exact sequence $0\to\ast_{j\geq{i+1}}M_{j}\to\ast_{j\geq i}M_{j}\to M_{i}\to 0$. 3. $(c)$

If $M_{n},M_{1}$ are regular, then $\ast_{j\geq i}M_{j}$ is regular.

Proof.

For $(c)$, just note that $M_{i}$ is balanced for $2\leq i\leq n-1$, hence $M_{i}$ is regular. ∎

Lemma 4.1.4**.**

Let $X,Y\in\operatorname{rep}(C_{r})$ be regular indecomposable. Then the following statments are equivalent.

1. $(a)$

There is $g\in G$ such that $\operatorname{Hom}(X^{g},Y)\neq 0$. 2. $(b)$

There is $h\in G$ such that $\operatorname{Hom}(\tau_{C_{r}}X^{h},\tau_{C_{r}}Y)$. 3. $(c)$

There is $l\in G$ such that $\operatorname{Hom}(\tau_{C_{r}}^{-1}X^{l},\tau_{C_{r}}^{-1}Y)$.

Proof.

We only show $(a)\Rightarrow(b)$. Let $g\in G$, such that $0\neq\operatorname{Hom}(X^{g},Y)$. Then

[TABLE]

Hence we find $h\in G$ such that $0\neq\operatorname{Hom}((\tau_{C_{r}}X)^{h},\tau_{C_{r}}Y)\cong\operatorname{Hom}(\tau_{C_{r}}X^{h},\tau_{C_{r}}Y)$. ∎

Proposition 4.1.5**.**

Let $n\geq 2$, $(M_{n},\ldots,M_{1})$ be leaf-connected and $M_{1},M_{n}$ regular, then

1. $(a)$

$\max\{d^{-}(\ast_{j\geq i}M_{j})\mid 1\leq i\leq n\}\leq d^{-}(\ast_{j\geq 1}M_{j})\leq\max\{d^{-}(M_{i})\mid 1\leq i\leq n\}$. 2. $(b)$

$\max\{d^{+}(\ast_{j\leq i}M_{j})\mid 1\leq i\leq n\}\leq d^{+}(\ast_{j\geq 1}M_{j})\leq\max\{d^{+}(M_{i})\mid 1\leq i\leq n\}$.

Proof.

Note that we have a filtration of $\ast_{j\geq 1}M_{j}$ by regular subrepresentations

[TABLE]

with $\ast_{j\geq i}M_{j}/\ast_{j\geq i+1}M_{j}\cong M_{i}$ regular for all $1\leq i\leq n$, where $\ast_{j\geq n+1}M_{j}:=0$.

$(a)$ Let $i\in\{1,\ldots,n\}$ and $Z:=\ast_{j\geq i}M_{i}$. Consider the short exact sequence

[TABLE]

Now $l\in\mathbb{Z}$ such that $\tau_{C_{r}}^{-l}Z\notin\operatorname{Inj}$. Then there exists $i\in\{1,\ldots,r\}$ and $g\in G$ such that $\operatorname{Hom}((X^{i})^{g},\tau_{C_{r}}^{-l}Z)\neq 0$. Hence we find $h\in G$ with $\operatorname{Hom}((\tau_{C_{r}}^{l}X^{i})^{h},Z)$ for some $h\in G$. Left-exactness of $\operatorname{Hom}((\tau_{C_{r}}^{l}X^{i})^{h},-)$ ensures that $0\neq\operatorname{Hom}((\tau_{C_{r}}^{l}X^{i})^{h},\ast_{j\geq 1}M_{j})$ and therefore we find $g\in G$ with $0\neq\operatorname{Hom}(X^{l}_{i},\tau_{C_{r}}^{-l}(\ast_{j\geq 1}M_{j}))$. Hence $\tau_{C_{r}}^{-l}(\ast_{j\geq 1}M_{j})\notin\operatorname{Inj}$ and therefore $d^{-}(X)\leq d^{-}(\ast_{j\geq 1}M_{j})$.

If $\operatorname{Hom}((X^{i})^{g},\tau_{C_{r}}^{-l}(\ast_{j\geq 1}M_{j}))\neq 0$ for some $g\in G$ and $i\in\{1,\ldots,r\}$, then we get find $h\in G$ with $0\neq\operatorname{Hom}(\tau_{C_{r}}^{l}(X^{i})^{h},\ast_{j\geq 1}M_{j})$. By [Ker3, 1.9] we find $1\leq p\leq n$ with $0\neq\operatorname{Hom}(\tau_{C_{r}}^{l}(X^{i})^{h},M_{p})$. Hence there is $g\in G$ with $0\neq\operatorname{Hom}((X^{i})^{g},\tau_{C_{r}}^{-l}M_{p})$. Hence $d^{-}(\ast_{j\geq 1}M_{j})\leq d^{-}(M_{p})$.

$(b)$ Note for $i\geq 2$ that $\ast_{j\geq 1}M_{j}/\ast_{j\geq i}M_{j}\cong\ast_{j<i}M_{j}$. Hence $D_{C_{r}}(\ast_{j\geq 1}M_{j})$ has a filtration

[TABLE]

Now apply $(a)$ and note that $d^{+}(X)=d^{-}(D_{C_{r}}X)$ for each regular indecomposable representation $X$. ∎

4.2. Small representations and trees

• Definition.

A balanced representation $N$ is called small if $1\leq d^{-}(N),d^{+}(N)\leq 2$.

Note that $N$ being balanced always implies $d^{-}(N)\geq 1$ and $d^{+}(N)\geq 1$.

• Definition.

Denote with $\overline{C_{r}}$ the underlying graph of $C_{r}$. Then $((C_{r})_{0},d)$ obtains the structure of a metric space, where $d(x,y)\in\mathbb{N}_{0}$ denotes the length of the unique path in $\overline{C_{r}}$ connecting vertices $x$ and $y$.

• Definition.

Let $T\subseteq C_{r}$ be a finite subtree. $T$ is called small if

1. $(a)$

$T$ has leaves in $C_{r}^{+}$ and $C_{r}^{-}$, 2. $(b)$

for all $x\in T_{0}$, we have $|T_{0}\cap n(x)|\leq 3$, 3. $(c)$

if $|T_{0}\cap n(x)|=3=|T_{0}\cap n(y)|$ then $x=y$ or $d(x,y)\geq 3$.

• Examples.

Let $l\in\mathbb{N}$ and $n\in 2\mathbb{N}$ with $n\geq 4l$. We denote with $\mathbb{A}_{l,n}\subseteq C_{r}$ a $($small$)$ subtree of the following form:

\bullet$$a_{1}$$\bullet$$a_{2}$$\bullet$$a_{3}$$\bullet$$a_{4}$$\bullet$$a_{5}$$\bullet$$a_{6}$$\bullet$$a_{7}$$\bullet$$a_{8}$$\bullet$$a_{9}$$\bullet$$\bullet$$\bullet$$t_{1}$$t_{2}$$t_{l}$$a_{4l-2}$$a_{4l-1}$$a_{n}…\bullet$$\bullet$$\bullet$$\bullet…\bullet$$\bullet$$\bullet$$\bullet

Lemma 4.2.1**.**

Assume $L$ is an indecomposable representation with small tree $T(L)$. Then $L$ is small.

Proof.

Since $T(L)$ is small with $T(L)_{0}=\operatorname{supp}(L)$, $L$ is balanced and therefore regular.

Let $l\in\mathbb{Z}$ be such that $\tau^{-l}_{C_{r}}L\notin\operatorname{Inj}$. By 3.5 and 4.1.4 we find $i\in\{1,\ldots,r\}$ and $g\in G$ with $0\neq\operatorname{Hom}((\tau_{C_{r}}^{l}X^{i})^{g},L)$. Fix $h\colon(\tau_{C_{r}}^{l}X^{i})^{g}\to L$ non-zero.

We assume that $l\geq 2$. By 3.6, $h$ is a monomorphism and $\operatorname{supp}((\tau_{C_{r}}^{l}X^{i})^{g})\subseteq\operatorname{supp}(L)$. Let $s$ be a sink of $(\tau_{C_{r}}^{l}X^{i})^{g}$. By 3.6, we have $|\operatorname{supp}(L)\cap n(s)|=r$. Since $T(L)$ is small we get $3\leq r=|\operatorname{supp}(L)\cap n(s)|\leq 3$. Hence

[TABLE]

Now let $t_{1},t_{2}\in\operatorname{supp}((\tau_{C_{r}}^{l}X^{i})^{g})$ be sinks. Then $(\ast)$ yields

[TABLE]

Since $T(L)$ is small we get $t_{1}=t_{2}$ or $d(t_{1},t_{2})\geq 3$. Since $C_{r}$ has bipartite orientation and $\operatorname{supp}((\tau_{C_{r}}^{l}X^{i})^{g})$ is connected, it follows $t_{1}=t_{2}$. Hence $\operatorname{supp}((\tau_{C_{r}}^{l}X^{i})^{g})$ contains exactly one sink $s$. Write $n(s)=\{a,b,c\}$. Since $l\geq 2$, $\ref{LemmaInjective}(a)$ implies $\operatorname{supp}((\tau_{C_{r}}^{l}X^{i})^{g})=\{s,a,b,c\}$. Hence $Z:=(\tau_{C_{r}}^{l}X^{i})^{g}$ can be considered as a representation of the Dynkin diagram $D_{4}$ with unique sink $s$ such that all linear maps are surjective. It follows that $Z_{x}=1$ for all $x\in\{s,a,b,c\}$. Hence $\operatorname{\underline{dim}}\pi_{\lambda}(Z)=(3,1)$ and $\pi_{\lambda}(Z)$ is indecomposable. But the only indecomposable representation $I\in\operatorname{rep}(\Gamma_{3})$ with dimension vector $(3,1)$ is injective. This is a contradiction since $Z$ is regular. Therefore $l\leq 1$ and $d^{-}(L)\leq 2$. For the other inequality note that $T(L)$ is small if and only if $T(D_{C_{r}}L)$ is small and $d^{+}(L)=d^{-}(D_{C_{r}}L)\leq 2$. ∎

5. The main theorem

Lemma 5.1**.**

Let $M\in\operatorname{rep}(\Gamma_{r})$ be an indecomposable representation with dimension vector $\operatorname{\underline{dim}}M=(a+1,a)$, $a\geq 1$. Then $M$ is a regular and quasi-simple representation.

Proof.

That $M$ is regular follows immediatly from [BoChen1, 2.1]. By [BoChen1, 3.4] it suffices to show that $A_{t}$ is not a common divisor of $a+1$ and $a$ for all $t\geq 2$, where $\operatorname{\underline{dim}}P_{i}=(A_{i-1},A_{i})$ is the dimension vector of the preprojective indecomposable representation $P_{i}$ $($see Figure $\ref{Fig:AuslanderReitenquiver})$. But this is trivial since $\operatorname{gcd}(a+1,a)=1$. ∎

Theorem 5.2**.**

Let $M\in\operatorname{rep}(C_{r})$ be balanced in the regular component $\mathcal{D}$ and $2\leq d^{+}(M),d^{-}(M)$. There is $n_{0}\in\mathbb{N}$ such that for each $n\geq n_{0}$ there is a regular component $\mathcal{D}_{n}$ with

[TABLE]

Moreover $\mathcal{D}_{n}$ contains a balanced quasi-simple representation $F_{n}$ with $\operatorname{\underline{dim}}\pi_{\lambda}(F_{n})=(n+1,n)$ or $\operatorname{\underline{dim}}\pi_{\lambda}(F_{n})=(n,n+1)$ and $D_{i}\neq D_{j}$ for $i\neq j\geq n_{0}$.

Proof.

Write $\operatorname{\underline{dim}}\pi_{\lambda}(M)=(a,b)$. After dualising $M$ we can assume $a\leq b$. Set $l:=2(b-a)+1\geq 1$ and $p_{0}:=4l$. Now let $p\geq p_{0}$ with $p\in 2\mathbb{N}$. Consider an indecomposable and thin representation $L$ $($i.e. $\dim_{k}L_{x}\leq 1$ for all $x\in(C_{r})_{0}$ such that $(L,M)$ is leaf-connected and $T(L)=T(\operatorname{supp}(L))$ is of type $\mathbb{A}_{l,p}$. Let $g\in G$ be such that $(M^{g},L)$ is leaf-connected. We conclude for $n:=2b+\frac{1}{2}p\in\mathbb{N}$, $n_{0}:=2b+\frac{1}{2}p_{0}$, and $F_{n}:=M^{g}\ast L\ast M$ that

[TABLE]

By 4.1.3 $F_{n}$ is a regular indecomposable representation and by 5.1 $\pi_{\lambda}(F_{n})$ is quasi-simple. Therefore $F_{n}$ is quasi-simple in a regular component $\mathcal{D}_{n}$. We conclude with 4.1.5

[TABLE]

i.e. $d^{-}(M)=d^{-}(F_{n})$. By the same token we have $d^{+}(M)=d^{+}(F_{n})$ and conclude

[TABLE]

It follows immediatly from the construction that the regular components are pairwise distinct, since $F_{i}$, $F_{j}$ are non-isomorphic and satisfy $\operatorname{ql}(F_{i})=\operatorname{ql}(F_{j})$ and $d^{-}(F_{i})=d^{-}(F_{j})$ for $i\neq j\geq n_{0}$. ∎

Corollary 5.3**.**

Let $M\in\mathcal{D}$ be balanced and $2\leq d^{+}(M),d^{-}(M)$. Then there exists a balanced and quasi-simple representation $F$ in a regular component $\mathcal{E}$ such that $\mathcal{W}_{C}(\mathcal{E})=\mathcal{W}_{C}(\mathcal{D})+\operatorname{ql}(M)-1$. Moreover there is a leaf $x\in C_{r}^{+}$ with $\dim_{k}F_{x}=1=\dim_{k}F_{y}$ for the unique element $y\in x^{+}\cap\operatorname{supp}(F)$.

Proof.

Fix $n\geq n_{0}$ in the proof of the theorem and set $F:=F_{n}=M^{g}\ast L\ast M$. The last claim follows since $L$ is a thin representation of type $\mathbb{A}_{l,p}$ which has $l+1\geq 2$ leaves in $C_{r}^{+}$ $($see Figure $\ref{Fig:ExampleSupport})$. ∎

6. Applications

6.1. Regular components for every width

The aim of this section is to construct for each $n\in\mathbb{N}$ a regular component $\mathcal{D}$ with $\mathcal{W}_{C}(\mathcal{D})=n$. Although each indecomposable representation has a leaf, it is in general not true that a regular representation has leaves in $C_{r}^{+}$ and $C_{r}^{-}$. For example if $M$ is indecomposable in $\operatorname{Inj}$, then each leaf of $M$ is a sink by 3.1. The next results shows that self-dual representations have leaves in $C_{r}^{+}$ and $C_{r}^{-}$.

Lemma 6.1.1**.**

Let $M\in\operatorname{rep}(C_{r})$ be indecomposable such that $D\pi_{\lambda}(M)\cong\pi_{\lambda}(M)$, then $M$ is balanced.

Proof.

Since $\operatorname{supp}(M)$ is finite there exists a leaf $x$ of $M$. Without loss of generality we assume that $x\in C_{r}^{+}$. We get $\pi_{\lambda}(M)\cong D\pi_{\lambda}(M)\cong\pi_{\lambda}(D_{C_{r}}M)$. Therefore we find $h\in G$ such that $M\cong(D_{C_{r}}M)^{h}.$ Since $h^{-1}.\varphi(x)\in C_{r}^{-}$ $($see $\ref{Duality})$ is a leaf of $(D_{C_{r}}M)^{h}$ the claim follows.∎

We denote with $\sigma C_{r}$ the quiver obtained by changing the orientation of all arrows in $C_{r}$. Note that $\sigma^{2}C_{r}=C_{r}$ and $\sigma C_{r}\cong C_{r}$. We denote by $\Phi^{+}$ the composition of the Bernstein-Gelfand-Ponomarev reflection functors [Assem1, VII 5.5.] for all the sources of $C_{r}$. $\Phi^{+}$ is a well-defined functor $\Phi^{+}\colon\operatorname{rep}(C_{r})\to\operatorname{rep}(\sigma C_{r})$ $($see [Ri6, 2.3]$)$. By the same token we have a functor $\Phi^{-}\colon\operatorname{rep}(\sigma C_{r})\to\operatorname{rep}(C_{r})$ given by the composition of the reflection functors for all the sources of $\sigma C_{r}$. Then $\mathcal{F}:=\Phi^{-}\circ\Phi^{+}\colon\operatorname{rep}(C_{r})\to\operatorname{rep}(C_{r})$ satisfies $\mathcal{F}(M)\cong\tau^{-1}_{C_{r}}M$ for $M\in\operatorname{rep}(C_{r})$ indecomposable and non-injective [Ri6, 2.3],[Assem1, VII 5.8.]. Therefore statements $(a)$ and $(b)$ of the next Lemma follow immediately from the definition of the reflection functors.

Lemma 6.1.2**.**

Let $M$ be in $\operatorname{rep}(C_{r})$ indecomposable and not injective.

1. $(a)$

For each $x\in C_{r}^{+}$ we have $\dim_{k}(\tau^{-1}_{C_{r}}M)_{x}=(\sum_{y\in x^{+}}\dim_{k}M_{y})-\dim_{k}M_{x}.$ 2. $(b)$

For each $y\in C_{r}^{-}$ we have $\dim_{k}(\tau^{-1}_{C_{r}}M)_{y}=(\sum_{x\in y^{-}}\dim_{k}(\tau^{-1}_{C_{r}}M)_{x})-\dim_{k}M_{y}.$ 3. $(c)$

Let $0\to A\to B\to C\to 0$ be an almost split sequence with $B$ indecomposable. If $a\in C^{-}_{r}$ is a leaf of $A$, then $B$ has a leaf in $C_{r}^{-}$. 4. $(d)$

Let $0\to A\to B\to C\to 0$ be an almost split sequence with $B$ indecomposable. If $a\in C_{r}^{+}$ is a leaf of $A$ and $b\in a^{+}$ satisfies $\dim_{k}A_{b}=\dim_{k}A_{a}$, then $a$ is a leaf of $B$.

Proof.

$(c)$ Consider a path $a\leftarrow b\to c$ such that $b,c$ are not in $\operatorname{supp}(A)$ as illustrated in Figure 6. Since $b$ is in $C_{r}^{+}$ we get with $(a)$ that

[TABLE]

Now let $d\in n(c)\setminus\{b\}$. Then $\dim_{k}C_{d}=(\sum_{y\in d^{+}}\dim_{k}A_{y})-\dim_{k}A_{d}=0$. Hence we get that $\dim_{k}C_{c}=\dim_{k}(\tau^{-1}_{C_{r}}A)_{c}\stackrel{{\scriptstyle(b)}}{{=}}(\sum_{x\in c^{-}}\dim_{k}(\tau^{-1}_{C_{r}}A)_{x})-\dim_{k}A_{c}=\dim_{k}C_{b}-\dim_{k}A_{c}=\dim_{k}C_{b}\neq 0.$ Hence $c\in\operatorname{supp}(C)$ is a leaf of $C$ and since $\operatorname{supp}(B)=\operatorname{supp}(A)\cup\operatorname{supp}(C)$ we conclude that $c\in C_{r}^{-}$ is a leaf of $B$.

$(d)$ Application of $(a)$ yields $($see Figure $\ref{Fig:ProofFigure2})$

[TABLE]

Now fix $c\in a^{+}\setminus\{b\}$, then $c\in C_{r}^{-}$. Let $d\in c^{-}\setminus\{a\}$ then $\dim_{k}A_{f}=0$ for all $f\in d^{+}\cup\{d\}$, since $f\notin\operatorname{supp}(A)$. Hence we get

[TABLE]

We conclude

[TABLE]

We have shown that $(a^{+}\setminus\{b\})\cap(\operatorname{supp}(A)\cup\operatorname{supp}(C))=\varnothing$. Since $\operatorname{supp}(A)\cup\operatorname{supp}(C)=\operatorname{supp}(B)$ we get $|\operatorname{supp}(B)\cap n(a)|=|(\operatorname{supp}(A)\cup\operatorname{supp}(C))\cap n(a)|\leq|\{b\}|=1$. Since $a\in\operatorname{supp}(A)\subseteq\operatorname{supp}(B)$ the vertex $a\in C_{r}^{+}$ is a leaf of $T(\operatorname{supp}(B))$. Since $B$ is indecomposable we have $T(B)=T(\operatorname{supp}(B))$.

∎

Let $M_{1}\in\operatorname{rep}(C_{r})$ be regular with $\dim_{k}M_{1}=2$. We define inductively a sequence of indecomposable representations in the regular component $\overline{\mathcal{D}}$ of $M_{1}$. The representation $M_{1}$ is quasi-simple. Assume that $M_{n}$ is already defined. If $n$ is odd, then $M_{n+1}$ is the unique indecomposable representation with irreducible epimorphism $M_{n+1}\to M_{n}$; if $n$ is even, then $M_{n+1}$ is the unique indecomposable representation with irreducible monomorphism $M_{n}\to M_{n+1}$. The component $\overline{\mathcal{D}}$ is shown in Figure 8. We have $\mathcal{W}_{C}(\overline{\mathcal{D}})=\mathcal{W}(\pi_{\lambda}(\overline{\mathcal{D}}))=1$ [Wor1, Example 3.3].

Theorem 6.1.3**.**

Let $r\geq 3$.

1. $(a)$

For each $m\geq 1$ there is $n_{0}\geq 1$ and a family $(\mathcal{D}_{n})_{n\geq n_{0}}$ of regular components with $\mathcal{W}_{C}(\mathcal{D}_{n})=m$ and $\mathcal{D}_{n}$ contains a quasi-simple representation $E_{n}$ with $\operatorname{\underline{dim}}\pi_{\lambda}(E_{n})=(n+1,n)$. 2. $(b)$

$\mathbb{N}\subseteq\{\mathcal{W}_{C}(\mathcal{E})\mid\mathcal{E}\in\mathcal{R}(C_{r})\}\subseteq\mathbb{N}_{0}$. 3. $(c)$

$\{\mathcal{W}(\mathcal{C})\mid\mathcal{C}\in\mathcal{R}\}=\mathbb{N}_{0}$.

Proof.

Recall that $\mathcal{W}_{C}(\overline{\mathcal{D}})=1$ and $\mathcal{W}_{C}(\overline{E})=2$ for the regular component $\overline{E}$ containing $X_{1}$. Fix $l\geq 1$. Then we have $D\pi_{\lambda}(M_{2l+1})\cong\pi_{\lambda}(M_{2l+1})$ by [Wor1, Example 3.3]. By 6.1.1 $M_{2l+1}$ is balanced. Moreover we have $d^{-}(M_{2l+1}),d^{+}(M_{2l+1})\geq 2$. Hence 5.2 yields $n_{0}\in\mathbb{N}$ and an infinite family of components $(\mathcal{D}_{n})_{n\geq n_{0}}$ of width $\mathcal{W}_{C}(\mathcal{D}_{n})=\mathcal{W}_{C}(\overline{\mathcal{D}})+\operatorname{ql}(M_{2l+1})-1=2l+1$. By Corollary 5.3 we find a balanced and quasi-simple representation $A\in\mathcal{D}_{n}$ that satisfies the assumption of $\ref{LemmaLeavesExistence}(d)$. Consider the AR sequence $0\to A\to B\to C\to 0$. Then $B$ is balanced by $\ref{LemmaLeavesExistence}(c),(d)$, $2\leq d^{+}(B),d^{-}(B)$ and $\operatorname{ql}(B)=2$. By 5.2 we get an infinite family of components of width $\mathcal{W}_{C}(\mathcal{D}_{n})+\operatorname{ql}(B)-1=(2l+1)+2-1=2l+2$. This proves $(a)$ and $(b)$. For $(c)$ observe that there exist regular components $\mathcal{C}\in\mathcal{R}$ with $\mathcal{W}(\mathcal{C})=0$ [Wor1, 3.3]. ∎

6.2. Counting regular components of fixed width

This section is motivated by the following result by Kerner and Lukas.

Proposition**.**

[Ker2, 5.2]* Assume that $k$ is uncountable and $A$ is a wild hereditary algebra with $n>2$ simple modules. Then the number of regular component of $A$ with quasi-rank $-1$ is uncountable. Moreover the set of components of quasi-rank $\leq-1$ for the Kronecker algebra is uncountable.*

The proof of the second statement uses the first statement for the path algebra $A$ of the wild quiver $1\rightarrow 2\rightrightarrows 3$ with $n(A)=3$ simple modules and the existence of a regular tilting module $T_{r}$ in $\operatorname{mod}A$ that induces a bijection

[TABLE]

with $\operatorname{rk}(\varphi(\mathcal{C}))\leq\operatorname{rk}(\mathcal{C})$ for all $\mathcal{C}\in\{\mathcal{D}\mid\mathcal{D}\ \text{regular component of}\ A\}$. To generalize the arguments from $\leq-1$ to $\leq-p$ for $p\in\mathbb{N}$ one would need the existence of bricks of arbitrary quasi-length. Unfortunately for each hereditary algebra there is a finite upper bound for the quasi-length of regular bricks given by the number of simple modules $-1$, which is in our case $n(A)-1=2$. We show how to circumvent this obstacle by considering an action of the general linear group $\operatorname{GL}_{r}(k)$ on $\operatorname{rep}(\Gamma_{r})$.

• Definition.

[CFP1, 3.6] Denote with $\operatorname{GL}_{r}(k)$ the group of invertible $r\times r$-matrices with coefficients in k which acts on $\bigoplus^{r}_{i=1}k\gamma_{i}$ via $A.\gamma_{j}=\sum^{r}_{i=1}a_{ij}\gamma_{i}$ for $1\leq j\leq r$, $A\in\operatorname{GL}_{r}(k)$. For $A\in\operatorname{GL}_{r}(k)$, let $\varphi_{A}\colon\mathcal{K}_{r}\to\mathcal{K}_{r}$ the algebra homomorphism with $\varphi_{A}(e_{1})=e_{1}$, $\varphi_{A}(e_{2})=e_{2}$ and $\varphi_{A}(\gamma_{i})=A.\gamma_{i}$, $1\leq i\leq r$. For a $\mathcal{K}_{r}$-module $M$ denote the pullback of $M$ along $\varphi_{A^{-1}}$ by $A.M$. The module $M$ is called $\operatorname{GL}_{r}(k)$-stable if $A.M\cong M$ for all $g\in\operatorname{GL}_{r}(k)$, in other words if $\operatorname{GL}_{r}=\operatorname{GL}_{r}(k)_{M}:=\{A\in\operatorname{GL}_{r}(k)\mid A.M\cong M\}$.

• Examples.

1. $(a)$

The simple representations of $\Gamma_{r}$ are $\operatorname{GL}_{r}(k)$-stable and by [Far1, 2.2] every preinjective and every preprojective representation is $\operatorname{GL}_{r}(k)$-stable. 2. $(b)$

There are $\operatorname{GL}_{r}(k)$-stable representations that are regular [Wor1, 1.2]. In this case all representations in the same component are also $\operatorname{GL}_{r}(k)$-stable. 3. $(c)$

Recall that the preinjective representation $I_{3}=\tau I_{1}$ has dimension vector $(3r-1,r)$. Let $M$ be in $\operatorname{rep}(C_{r})$ with $\pi_{\lambda}(M)\cong I_{3}$. The support of $M$ for $r=3$ is shown in Figure 9. Let $c\in T(M)_{0}$ be the vertex $\dim_{k}M_{c}=2$.

The underlying tree of $\operatorname{supp}(M)$ is symmetric in the following sense. The quiver $T(M)\setminus\{c\}$ is not connected and consists of $r=3$ isomorphic trees $T_{1},T_{2},T_{3}$. Moreover for $n\in\{1,2\}$ the sum $\dim_{k}M_{x}$ for $x\in(T_{i})_{0}$ with distance $d(c,x)=n$ is independent of $i\in\{1,2,3\}$. We will prove that this is not a coincidence. We show that every representation $M$ such that $\pi_{\lambda}(M)$ is $\operatorname{GL}_{r}(k)$-stable, has a central point.

6.2.1. $\mathcal{S}_{r}$-stability

Denote by $\mathcal{S}_{r}$ the symmetric group on $\{1,\ldots,r\}$. Then each for each $\sigma\in\mathcal{S}_{r}$ there is an induced bijection on $\{1,\ldots,r\}\to(\Gamma_{r})_{1}$ given by $i\mapsto\gamma_{\sigma(i)}$ which extends in a natural way to the set of equivalence classes of walks in $\Gamma_{r}$. By abuse of notation we denote by $\sigma\colon C_{r}\to C_{r}$ the induced quiver automorphism. Let $\alpha\colon[w]\to[\gamma_{i}w]$ be an arrow in $C_{r}$, then by definition $\sigma(\alpha)$ is the unique arrow $\sigma(\alpha)\colon\sigma([w])\to[\gamma_{\sigma(i)}]\sigma([w])$. Note that $\pi(\alpha)=\gamma_{i}$ and $\pi(\sigma(\alpha))=\gamma_{\sigma(i)}$.

Now let $M\in\operatorname{rep}(C_{r})$ be an indecomposable representation. We define $\sigma(M)$ to be the indecomposable representation with

[TABLE]

We say that $M$ is $\mathcal{S}_{r}$-stable if for each $\sigma\in\mathcal{S}_{r}$ there is $g_{\sigma}$ with $M\cong\sigma(M)^{g_{\sigma}}$. This definition is motivated by the following obvious result:

Corollary 6.2.1**.**

Let $M\in\operatorname{rep}(C_{r})$ be an indecomposable representation. If $\pi_{\lambda}(M)$ is $\operatorname{GL}_{r}(k)$-stable then $M$ is $\mathcal{S}_{r}$-stable.

Proof.

We let $I(\sigma)$ be the permutation matrix given by $\sigma$, i.e. $I(\sigma)_{ij}=1$ if and only if $\sigma(i)=j$ and $I(\sigma)_{ij}=0$ otherwise. Now we assume that $\pi_{\lambda}(M)$ is $\operatorname{GL}_{r}(k)$-stable. Then we get for each $\sigma\in\mathcal{S}_{r}$ that

[TABLE]

Hence we find $g_{\sigma}\in G$ such that

[TABLE]

∎

Note that $\sigma([\epsilon_{1}])=[\epsilon_{1}]$ and since $G$ acts freely $C_{r}$, the element $g_{\sigma}$ is uniquely determined. In the following we study the quiver automorphisms $\sigma\circ g_{\sigma}\colon C_{r}\to C_{r}$.

6.2.2. Automorphisms of trees

A group G is said to have property FA [Se1, I.6.1] if every action of G on a tree $T$ by graph automorphisms (which do not invert an edge) has a global fixed point $z\in T_{0}$, i.e. $gz=z$ for all $g\in G$. It is known [Se1, I.6.3.1] that all finitely generated torsion groups have the property FA. In particular, for every group action of a finite group acting on a quiver which underlying graph is a tree, there is a global fixed point.

• Definition.

For $x\in C_{r}^{+}$ and $1\leq i\leq r$ denote by $T(x,i)$ the connected component of $C_{r}\setminus\{x\}$ containing $t(\alpha_{i})$, where $\alpha_{i}\colon x\to t(\alpha_{i})$ is the unique arrow with $\pi(\alpha_{i})=\gamma_{i}$.

Let $M$ in $\operatorname{rep}(C_{r})$ be indecomposable and $x\in\operatorname{supp}(M)$, then we define $T(x,i,M):=T(M)\cap T(x,i)$.

Note that $\operatorname{supp}(M)=\{x\}\cup\bigcup^{r}_{i=1}T(x,i,M)_{0}$.

Proposition 6.2.2**.**

Let $M\in\operatorname{rep}(C_{r})$ be $\mathcal{S}_{r}$-stable. Then there is $c\in\operatorname{supp}(M)$ such that

1. $(a)$

$\sigma\circ g_{\sigma}(c)=c$* for all $\sigma\in\mathcal{S}_{r}$.* 2. $(b)$

For each $n\in\mathbb{N}$ the number $r$ divides $D(n,c):=\sum_{x\in\operatorname{supp}(M),d(x,c)=n}M_{x}$.

Proof.

Since $M\cong\sigma(M)^{g_{\sigma}}$, we have

[TABLE]

We assume that $\dim_{k}M\neq 1$, otherwise there is nothing to show. Let $T\subseteq C_{r}$ be the finite subtree $T:=T(M)$. Then $\sigma\circ g_{\sigma}\colon T\to T$ is a quiver automorphism of $T$. Since $T$ is finite, $\operatorname{Aut}(T)$ is finite and there exists a vertex $c\in T_{0}$ with $\varphi(c)=c$ for all $\varphi\in\operatorname{Aut}(T)$. We assume that $c\in C_{r}^{+}$. For $1\leq i\leq r$ we let $\beta_{i}\colon c\to t(\beta_{i})$ be the unique arrow with $\pi(\beta_{i})=\gamma_{i}$ and set $T_{i}:=T(c,i,M)$. Since $\dim_{k}M\neq 1$ and $M$ is indecomposable, we can assume w.l.o.g. that $e:=t(\beta_{1})\in\operatorname{supp}(M)$. Since every automorphism of $C_{r}$ respects the metric $($see $\ref{DefinitionMetric})$ we get

[TABLE]

In particular, $\sigma\circ g_{\sigma}(\beta_{1})\in\{\beta_{1},\ldots,\beta_{r}\}$. Now fix $j\in\{1,\ldots,r\}\setminus\{1\}$ and $\sigma:=(1\ j)\in\mathcal{S}_{r}$. Then we have $\pi(\sigma(\beta_{1}))=\gamma_{j}$. Since $\pi\circ g=\pi$ for all $g\in G$, we get $\sigma\circ g_{\sigma}(\beta_{1})=\beta_{j}$ and conclude $\sigma\circ g_{\sigma}(T_{1})=T_{j}$. Hence $T_{1},\ldots,T_{r}$ are non-empty isomorphic quivers. For each $n\in\mathbb{N}$ and $i\in\{1,\ldots,r\}$ we define $d_{n,i,c}:=\{x\in(T_{i})_{0}\mid d(x,c)=n\}$. Let now $x\in d_{n,1,c}$, then we have $\sigma\circ g_{\sigma}(x)\in d_{n,j,c}$ since $\sigma\circ g_{\sigma}(x)\in T_{j}$ and

[TABLE]

Moreover we have $M_{x}=(\sigma(M)^{g_{\sigma}})_{x}=M_{\sigma\circ g_{\sigma}(x)}$. It follows $\sum_{y\in d_{n,1,c}}\dim_{k}M_{y}=\sum_{y\in d_{n,j,c}}\dim_{k}M_{y}$ and we conclude

[TABLE]

∎

Corollary 6.2.3**.**

Assume that $M$ is $\mathcal{S}_{r}$-stable. If $\dim\pi_{\lambda}(M)=(a,b)$ then $r$ divides $a$ or $b$.

Proof.

Let $c\in\operatorname{supp}(M)$ be as in 6.2.2. Then we have

[TABLE]

If $c\in C^{+}_{r}$, then $b=\sum_{n\in 2\mathbb{N}-1}D(n,c)$ and $a=\sum_{n\in 2\mathbb{N}-1}D(n,c)$ otherwise. Hence $r$ divides $b$ or $a$. ∎

As an application we get the following result for components of the Kronecker quiver $\Gamma_{r}$.

Corollary 6.2.4**.**

Let $m\in\mathbb{N}$, then there exists a regular component $\mathcal{C}$ with $\mathcal{W}(\mathcal{C})=m$ and no representation in $\mathcal{C}$ is $\operatorname{GL}_{r}(k)$-stable.

Proof.

Let $m\geq 1$. By 5.2 there exists $n_{0}\in\mathbb{N}$ such that for each $n\geq n_{0}$ there is a regular component $\mathcal{D}_{n}$ with $\mathcal{W}(\mathcal{D}_{n})=m$ and $\mathcal{D}_{m}$ contains a quasi-simple representation $E_{n}=\pi_{\lambda}(F_{n})$ with $F_{n}\in\operatorname{rep}(C_{r})$ and $\operatorname{\underline{dim}}E_{n}=(n+1,n)$. Since $r\geq 3$, we find $l\geq n_{0}$ $($even infinitely many$)$ such that $r$ does not divide $l$ and $l+1$. Hence $F_{l}$ is not $\mathcal{S}_{r}$-stable and $E_{l}$ not $\operatorname{GL}_{r}(k)$-stable. Therefore no representation in $\mathcal{D}_{l}$ is $\operatorname{GL}_{r}(k)$-stable by [Far1, 2.2]. ∎

6.2.3. The number of regular components in $\operatorname{rep}(\Gamma_{r})$

• Definition.

A locally closed set is an open subset of a closed set. A constructible set is a finite union of locally closed sets.

Lemma 6.2.5**.**

Let $M\in\operatorname{rep}(\Gamma_{r})$ with $\operatorname{GL}_{r}(k)_{M}\neq\operatorname{GL}_{r}(k)$. There is an injection $\iota\colon k\to\operatorname{GL}_{r}(k)/\operatorname{GL}_{r}(k)_{M}$.

Proof.

By [Far1, 2.1] $\operatorname{GL}_{r}(k)_{M}$ is a closed subgroup of $\operatorname{GL}_{r}(k)$ and by [Mall, 5.5] an $\operatorname{GL}_{r}(k)/\operatorname{GL}_{r}(k)_{M}$ an algebraic variety. Hence we find an affine variety $V\subseteq\operatorname{GL}_{r}(k)/\operatorname{GL}_{r}(k)_{M}$ with $d:=\dim V=\dim\operatorname{GL}_{r}(k)/\operatorname{GL}_{r}(k)_{M}$. Since $\operatorname{GL}_{r}(k)$ is irreducible we have $\dim\operatorname{GL}_{r}(k)/\operatorname{GL}_{r}(k)_{M}=\dim\operatorname{GL}_{r}(k)-\dim\operatorname{GL}_{r}(k)_{M}\geq 1$. Let $k[t_{1},\ldots,t_{d}]\to k[V]$ be a Noether-normalization and $\varphi^{\ast}\colon V\to\mathbb{A}^{d}$ be the comorphism. Then $\varphi^{\ast}$ is dominant. Hence there is a dominant morphism $f\colon V\to\mathbb{A}^{1}$. By Chevalley’s Theorem $f(V)$ is constructible and hence finite or cofinite. Since $f(V)$ is dense in $\mathbb{A}$, $f(V)$ is not finite and therefore cofinite. That means $|k\setminus C|$ is finite. Since $k$ is infinite we have $|f(V)|=|k|$. It follows $|k|=|\mathbb{A}|=|f(V)|\leq|V|\leq|\operatorname{GL}_{r}(M)/\operatorname{GL}_{r}(k)_{M}|.$ ∎

Theorem 6.2.6**.**

Let $m\in\mathbb{N}$. There is a bijection $\{\mathcal{C}\in\mathcal{R}\mid\mathcal{W}(\mathcal{C})=m\}\to k$.

Proof.

It is well known that $|\mathcal{R}|=|k|$, see [Assem3, XVIII 1.8]. In particular there is an injection

[TABLE]

By 6.2.4 there is a regular component $\mathcal{C}$ with $\mathcal{W}(\mathcal{C})=m\in\mathbb{N}$ such that no representation in $\mathcal{C}$ is $\operatorname{GL}_{r}(k)$-stable. For $E$ in $\mathcal{C}$ the map

[TABLE]

is well defined and injective. Since the number of representations in a regular component with given dimension vector $(a,b)$ is $\leq 1$ [Assem3, XIII.1.7] and $\operatorname{GL}_{r}(k)$ acts via auto equivalances we get with 6.2.5 an injection

[TABLE]

By the Schröder-Bernstein Theorem we get the desired bijection. ∎

• Remark.

Note that we restrict ourselves to components of width $\geq 1$, since we dont know whether components in $\operatorname{rep}(C_{r})$ of width [math] exist. Also the examples [Wor1, 3.3] of components of width [math] in $\operatorname{rep}(\Gamma_{r})$ are $\operatorname{GL}_{r}(k)$-stable. For the case $n=0$ we argue as follows.

Lemma 6.2.7**.**

Let $r\geq 3$, then there exists an bijection $k\to\{\mathcal{C}\in\mathcal{R}\mid\mathcal{W}(\mathcal{C})=0\}$.

Proof.

The proof of [Bi1, 3.3.3] yields an injective map $\varphi\colon\operatorname{ind}\mathcal{E}(\mathcal{X})\to\{\mathcal{C}\in\mathcal{R}\mid\mathcal{W}(\mathcal{C})=0\}$, where $\operatorname{ind}\mathcal{E}(\mathcal{X})$ are the indecomposable objects in a category $\mathcal{E}(\mathcal{X})$ equivalent to the category of finite dimensional modules over the power-series ring $k\langle\langle X_{1},\ldots,X_{t}\rangle\rangle$ in non-commuting variables $X_{1},\ldots,X_{t}$ and $t\geq 2$. Now let $\lambda\in k\setminus\{0\}$ and consider the indecomposable module $M_{\lambda}=k^{2}$ given by $X_{1}.(a,b)=(\lambda b,0)$, $X_{2}.(a,b)=(b,0)$ and $X_{i}.(a,b)=0$ for $i>2$. Then $M_{\lambda}\not\cong M_{\mu}$ for $\lambda\neq\mu$ and we have an injection $k\to\operatorname{ind}\mathcal{E}(\mathcal{X})$. The claim follows as in 6.2.6. ∎

Corollary 6.2.8**.**

Let $r\geq 3$, then for each $n\in\mathbb{N}$ there are exactly $|k|$ regular components such that $\operatorname{rk}(\mathcal{C})\in[-n,-n+3]$.

Corollary 6.2.9**.**

Assume that $k$ is uncountable and $p\in\mathbb{N}$. The set of components of quasi-rank $\leq-p$ in $\mathcal{R}$ is uncountable.

Acknowledgement

The results of this article are part of my doctoral thesis, which I am currently writing at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for his continuous support and helpful comments. I also would like to thank the whole research team for the very pleasant working atmosphere and the encouragement throughout my studies. In particular, I thank Christian Drenkhahn for proofreading.

Furthermore, I thank Claus Michael Ringel for fruitful discussions during my visits in Bielefeld.