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\title{Reflection Functors in the Representation Theory of Quivers}
\author{Danika Van Niel}

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$\textbf{Abstract:}$
The paper "Coxeter Functors and Gabriel's Theorem" written by I.N. Bernstein, I.M. Gel'fand, and V.A. Ponomarev explores the concept of reflection functors. A thorough proof of several results used by Bernstein et al in their paper is presented. The focus is on the category of representations and reflection functors, both negative and positive. The quadratic form is the bridge between the results on quivers and the techniques of Lie algebras. The Dynkin diagrams mentioned in Gabriel's Theorem are discussed.
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$\textbf{Executive Summary:}$

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The purpose of this paper is to thoroughly prove some of the important results that are used in the paper "Coxeter Functors and Gabriel's Theorem" by Bernstein et al [1]. The focus is mostly on the category of representations and the reflection functors to better understand how they can be used to prove Gabriel's Theorem. Gabriel's Theorem was initially not proved through Lie algebra or representation theory but it gave results about the Dynkin Diagrams which were previously only related to those two fields. Bernstein et al wrote another proof of Gabriel's Theorem using tools from representation theory, namely the reflection functors. This offers a relation between these fields of mathematics.

Consider a graph which is a set of a finite number of vertices and edges, namely $\Gamma$. Then we place an orientation on it which makes the edges arrows so that they have an orientation, namely $\Lambda$. The category $\mathscr L (\Gamma, \Lambda)$ has objects and morphisms. Objects are collections of vector spaces and linear mappings which go between the vector spaces. Morphisms are a logical way to compare objects.

We showed that $\mathscr L$$(\Gamma,\Lambda) satisfies the following conditions and therefore is a category: \begin{enumerate} \item The composition of morphisms is a morphism and the composition is associative \item For all morphisms \phi: (U,f) \to (V,g), \, 1_{(V,g)}\phi = \phi 1_{(U,g)} = \phi \end{enumerate} Reflection functors change representations. For example look at an orientation \Lambda where there is a vertex \beta such that all of the arrows that are connected to \beta are going into the vertex (referred to as a sink), then F_\beta^+ (referred to as a positive reflection functor) changes \mathscr L (\Gamma, \Lambda) to \mathscr L (\Gamma, \sigma_\beta \Lambda) where \sigma_\beta \Lambda looks exactly like \Lambda except that instead of all of the arrows going into \beta all of the arrows are coming out of \beta (referred to as a source). The vertices are vector spaces and the arrows are linear mappings, therefore since the vertices don't change between \Lambda and \sigma_\beta \Lambda, but the arrows do then the vector spaces don't change and the linear mappings do. Therefore we must check that how we defined the reflection functors, both positive and negative for a sink and a source respectively, work properly. We show that F_\beta^+: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\beta\Lambda) satisfies the following conditions and therefore is a functor: \begin{enumerate} \item F_\beta^+ (1_{(U,f)}) = 1_{(X,r)} \item F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi)) \end{enumerate} Similarly we can show that F_\alpha^- is a functor. After proving that F_\beta^+ and F_\alpha^- are both functors, we can now use Theorem 1, and Lemma 1. We use statements and mappings that we used earlier to prove the Theorem 1 and Lemma 1. From the Theorem and Lemma we can immediatly prove Corollary 1. These proofs give us more insight in how the functors can be used, and what properites that they have in a more abstract way. We discuss the quadratic form in order to bridge the relationship between the results on quivers and the techniques of Lie algebras. This brings us closer to our goal of abstractly showing how these different fields of mathematics are related. Now to show the main idea of this paper we will show how the reflection functors F^+_\beta and F^-_\alpha were used to prove part 2 of the famous Gabriel's Theorem. This is not the first way that Gabriel's Theorem was proven, therefore the two fields of mathematics which the two different proofs came from are connected in this way. \newpage \cfoot{} \lfoot{\thepage} \rfoot{} \renewcommand{\headrulewidth}{0pt} \tableofcontents \newpage \cfoot{} \rfoot{\thepage} \lfoot{} \section{Introduction} This project is about representations of quivers which is an area of mathematics that uses methods of linear algebra, combinatorics and category theory. \\ Recall some necessary definitions from linear algebra. \\ \indent Let V and W be vector spaces over a fixed field K. A function \psi: V \to W is a \textbf{linear mapping} if \psi(u+v) = \psi(u) + \psi(v) and \psi(cu) = c\psi(u) for all u,v \in V and c \in K. If \phi: U \to V is another linear mapping, then the composition \psi \circ \phi: U \to W is defined by [\psi \circ \phi](u) = \psi(\phi(u)). Sometimes we write \psi\phi instead of \psi \circ \phi. The following two definitions are from the text Homology by Saunders Mac Lane. The \textbf{kernel} of a morphism h: V \to W, Ker\,\psi, consists of all v \in V such that \psi(v) = 0. The following is a universal property: for each \phi: U \to V satisfying \psi \phi = 0, there exists a unique \xi: U \to Ker\,\psi with \phi = \kappa \xi, \kappa the inclusion map. \centerline{ \xymatrix{ Ker\,\psi \ar[r]^\kappa & V \ar[r]^\psi & W \\ U \ar[u]^\xi \ar[ru]_\phi } } \noindent The \textbf{cokernel} of a morphism \widetilde{h}: V \to W, Coker\,\widetilde{h}, is equal to the quotient module W/Im\,\widetilde{h}. The following is a universal property: for each \phi: W \to U satisfying \phi\psi = 0, there exists a unique \xi: Coker\,\psi \to U with \phi = \xi\pi, \pi the natural projection map. \centerline{ \xymatrix{ V \ar[r]^{\psi} & W \ar[r]^{\pi} \ar[dr]_{\phi} & Coker\,\psi \ar[d]^{\xi} \\ && U } } \noindent The \textbf{identity mapping} 1_U:U \to U is given by 1_U(u) = u for all u \in U. We use the fact that the composition of linear mappings is associative, i.e. if \phi and \psi are as above and \xi: W \to Y is a linear mapping, then (\xi \, \circ \, \psi) \circ \phi = \xi \circ (\psi \, \circ \, \phi). We also use the fact that 1_V \circ \phi = \phi \circ 1_U = \phi for all \phi as above. Recall that the vector space V is finite dimensional if it has a finite spanning set. \par A linear map \psi: V \to W is an isomporhpism if there exists a linear map \zeta: W \to V satisfying \psi \, \circ \, \zeta = 1_W and \zeta \, \circ \, \psi = 1_V. It is a standard fact that a linear map is an isomorphism if and only if it is both injective and surjective. Vector spaces V and W are isomorphic if there exists an isomporphism V \to W. \par If V and W are vector spaces, the direct sum V \oplus W is the set of all pairs (v,w) such that v \in V and w \in W with component-wise addition and scalar multiplication. If \mu: V \to V' and \nu: W \to W' are linear maps, then the direct sum \mu \oplus \nu: V \oplus W \to V' \oplus W' is defined by (\mu \oplus \nu) (v,w) = (\nu(v), \mu(w)). If \phi: V' \to V'', \psi: W' \to W'' are linear maps, then (\phi \oplus \psi)(\mu \oplus \nu) = \phi \mu \oplus \psi \nu. A categorical definition of a direct sum is that a vector space X is isomorphic to V \oplus W if and only if there exist four linear maps V \underset{\pi_V}{\stackrel{\iota_V}{\rightleftarrows}} X \underset{\pi_W}{\stackrel{\iota_W}{\leftrightarrows}} W satisfying \pi_V\iota_V = 1_V, \pi_W\iota_W = 1_W, and \iota_V\pi_V + \iota_W\pi_W = 1_X. In the special case when X = V \oplus W as above then the maps are defined as follows: \iota_V: V \to X, \iota_W: W \to X, \pi_V: X \to V, and \pi_W: X \to W such that \iota(v) = (v, 0), \iota(w) = (0, w), \pi_V(v,w) = v, and \pi_W(v,w) = w where v \in V, w \in W, and (v, w) \in X. \par We present the following facts from the "Coxeter Functors and Gabriel's Theorem" paper written by I.N. Bernstein, I.M. Gel'fand, and V.A. Ponomarev. Define \Gamma as a finite connected graph with the set of vertices \Gamma_0 and the set of edges \Gamma_1. Fix an \newpage \cfoot{} \lfoot{\thepage} \rfoot{} \noindent orientation \Lambda of the graph \Gamma which assigns to each edge \ell \in \Gamma_1 a starting point \alpha(\ell) \in \Gamma_0 and an end-point \beta(\ell) \in \Gamma_0. We obtain a directed (oriented) graph which we call a quiver and denote by (\Gamma, \Lambda). With the reference to a general definition of a category in Homology by Saunders Mac Lane we define a \textbf{category} \mathscr L$$(\Gamma,\Lambda)$ as follows. A category consists of objects and morphisms which may sometimes be composed. An object of $\mathscr L$$(\Gamma,\Lambda) is any collection (V,f) of finite dimensional vector spaces V_\alpha \, (\alpha \in \Gamma_0) and linear mappings f_\ell (\ell \in \Gamma_1). There is a particular representation where all the vector spaces are zero and all the maps are the zero maps, called 0. A \textbf{morphism} \phi: (V,f) \to (W,g) is a collection of linear mappings \phi_\alpha: V_\alpha \to W_\alpha (\alpha \in \Gamma_0) such that for each edge \ell \in \Gamma_1 the following diagram \centerline{ \xymatrix{ V_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]^{\phi_{\alpha(\ell)}} & V_{\beta(\ell)} \ar[d]^{\phi_{\beta(\ell)}} \\ W_{\alpha(\ell)} \ar[r]_{g_\ell} & W_{\beta(\ell)} } } \noindent is commutative, that is, \phi_{\beta(\ell)} f_\ell = g_\ell \phi_{\alpha(\ell)}. The objects of \mathscr L$$(\Gamma,\Lambda)$ are called representations of the quiver $(\Gamma,\Lambda)$ and the category $\mathscr L$$(\Gamma,\Lambda) is called the category of representations of (\Gamma,\Lambda). \par We define the law of composition for morphisms as follows. Let \phi: (U,f) \to (V,g) and \psi: (V,g) \to (W,h) be morphisms where \phi = (\phi_\alpha)_{\alpha \in \Gamma_0} and \psi = (\psi_\alpha)_{\alpha \in \Gamma_0}. Then \psi \circ \phi: (U,f) \to (W,h) is given by (\psi \circ \phi)_\alpha = \psi_\alpha \circ \phi_\alpha. \\[11pt] \noindent Define the \textbf{identity morphism} 1_{(V,f)} for an object (V,f) by 1_{(V,f)} = (1_{V_\alpha})_{\alpha \in \Gamma_0}. We prove that \mathscr L$$(\Gamma,\Lambda)$ is a category in the next section.

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\section{The Category of Representations}

\noindent We show that $\mathscr L$$(\Gamma,\Lambda) satisfies the following conditions and therefore is a category: \begin{enumerate} \item The composition of morphisms is a morphism and the composition is associative \item For all morphisms \phi: (U,f) \to (V,g), \, 1_{(V,g)}\phi = \phi 1_{(U,g)} = \phi \end{enumerate} For any objects (U,f), (V,g), and (W,h) in \mathscr L$$(\Gamma,\Lambda)$, let $\phi: (U,f) \to (V,g)$ and \\ $\psi: (V,g) \to (W,h)$ be morphisms. Then we have a commutative diagram,

\centerline{
\xymatrix{
U_{\alpha(\ell)} \ar[d]_{\phi_{\alpha(\ell)}} \ar[r]^{f_\ell} & U_{\beta(\ell)} \ar[d]^{\phi_{\beta(\ell)}} \\
V_{\alpha(\ell)} \ar[d]_{\psi_{\alpha(\ell)}} \ar[r]^{g_\ell} & V_{\beta(\ell)} \ar[d]^{\psi_{\beta(\ell)}} \\
W_{\alpha(\ell)} \ar[r]^{h_\ell} & W_{\beta(\ell)}
}
}

\noindent that is
\begin{equation} \label{1}
\phi_{\beta(\ell)}f_\ell = g_\ell\phi_{\alpha(\ell)}
\end{equation}
and $\psi_{\beta(\ell)} g_\ell = h_\ell \psi_{\alpha(\ell)}$. Then

\begin{center}
$\psi_{\beta(\ell)} \phi_{\beta(\ell)} f_\ell = \psi_{\beta(\ell)} (\phi_{\beta(\ell)} f_\ell) = \psi_{\beta(\ell)} (g_\ell \phi_{\alpha(\ell)}) =$ \\
$(\psi_{\beta(\ell)} g_\ell) \phi_{\alpha(\ell)} = (h_\ell \psi_{\alpha(\ell)}) \phi_{\alpha(\ell)} = h_\ell \psi_{\alpha(\ell)} \phi_{\alpha(\ell)}$
\end{center}
\noindent which shows that $\psi \circ \phi : (U,f) \to (W,h)$ is a morphism, that is the diagram

\centerline{
\xymatrix{
U_{\alpha(\ell)} \ar[d]_{[\psi \circ \phi]_{\alpha(\ell)}} \ar[r]^{f_\ell} & U_{\beta(\ell)} \ar[d]^{[\psi \circ \phi]_{\beta(\ell)}} \\
W_{\alpha(\ell)} \ar[r]^{h_\ell} & W_{\beta(\ell)}
}
}

\noindent commutes.
\\[12pt]
We have shown that the composition of morphisms is well-defined.
\\[12pt]
Suppose that $\phi$ and $\psi$ are as above, and $\xi: (W,h) \to (Y,j)$ is a morphism in $\mathscr L$$(\Gamma,\Lambda) where \xi = (\xi_\alpha), \, \alpha \in \Gamma_0 Then, using the associativity of composition of linear mappings we get \begin{center} [(\xi \circ \psi) \circ \phi]_\alpha = (\xi \circ \psi)_\alpha \circ \phi_\alpha = (\xi_\alpha \circ \psi_\alpha) \circ \phi_\alpha = \xi_\alpha \circ (\psi_\alpha \circ \phi_\alpha) = \xi_\alpha \circ (\psi \circ \phi)_\alpha = [\xi \circ (\psi \circ \phi)]_\alpha. \end{center} Therefore, (\xi \circ \psi) \circ \phi = \xi \circ (\psi \circ \phi). We have shown the composition of morphisms is associative. Thus \mathscr L$$(\Gamma,\Lambda)$ satisfies the first property.
\\[12pt]
For a morphism $\phi : (U,f) \to (V,g)$ as above, we have

\begin{center}
$[1_{(V,g)} \circ \phi]_\alpha = (1_{(V,g)})_\alpha \circ \phi_\alpha = \phi_\alpha$ \, and \,
$[\phi \circ 1_{(U,f)}]_\alpha = \phi_\alpha \circ (1_{(U,f)})_\alpha = \phi_\alpha$.
\end{center}

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\noindent Therefore $1_{(V,g)} \circ \phi = \phi \circ 1_{(U,f)} = \phi$. We have shown that $\mathscr L$$(\Gamma,\Lambda) satisfies the second property. We have shown that all of the axioms of a category defined in Homology by Saunders Mac Lane meaning that we have shown \mathscr L$$(\Gamma,\Lambda)$ is a category.

\par A morphism $\psi: (V,g) \to (W,h)$ is an isomorphism if there exists a morphism $\zeta: (W,h) \to (V,g)$ satisfying $\psi \, \circ \, \zeta = 1_{(W,h)}$ and $\zeta \, \circ \, \psi = 1_{(V,g)}$. Representations of quivers $(V,g)$ and $(W,h)$ of the quiver $(\Gamma, \Lambda)$ are isomorphic if there exists an isomorhpism $(V,g) \to (W,h)$. If $(V, g)$, $(W,h)$ are representations then the set of morphisms $(V,g) \to (W,h)$ is a finite dimensional vector space over the field $K$. \\
\centerline{
$\phi = (\phi_\alpha)_{\alpha \in \Gamma_0} \, , \, \psi = (\psi_\alpha)_{\alpha \in \Gamma_0}$}
We define $\phi + \psi$ by

\centerline{
$(\phi + \psi)_\alpha = \phi_\alpha + \psi_\alpha$}

\noindent and, for $c \in K$ we define $c\phi$ by

\centerline{
$(c\phi)_\alpha = c\phi_\alpha$.
}

Referencing Equation (\ref{1}) we have $\phi_{\beta(\ell)}f_\ell = g_\ell\phi_{\alpha(\ell)}$ and $\psi_{\beta(\ell)}f_\ell = g_\ell\psi_{\alpha(\ell)}$. Adding the left hand sides and right hand sides gives us $(\phi_{\beta(\ell)}+ \psi_{\beta(\ell)})f_\ell = g_\ell(\phi_{\alpha(\ell)} + \psi_{\alpha(\ell)})$ which shows $\phi + \psi$ is a morphism.

The verification that $c\phi$ is a morphism is similar.

In view of our definition of the sums of the morphisms, and the scalar multiplication, the above verification also shows that Hom$_{\mathscr L(\Gamma,\Lambda)}((U,f),(V,g))$ $\subset$ $\underset{\alpha \in \Gamma_0}{\oplus}$ Hom$_K(U_\alpha, V_\alpha)$ is a subspace. Therefore since we know that $\underset{\alpha \in \Gamma_0}{\oplus}$ Hom$_K(U_\alpha, V_\alpha)$ is finite dimensional, then \\ Hom$_{\mathscr L(\Gamma,\Lambda)}((U,f),(V,g))$ is finite dimensional.

A verification similar to above shows that $\phi(\psi + \xi) = \phi\psi + \phi\xi$ and $(\phi +\xi)\psi = \phi\psi + \xi\psi$ is true for $\mathscr L(\Gamma,\Lambda)$, therefore we know that $\mathscr L(\Gamma,\Lambda)$ is a preadditive. It is easy to verify that $c(\phi\psi) = (c\phi)\psi = \phi(c\psi)$ so $\mathscr L(\Gamma,\Lambda)$ is a $k$-category.

\par If $(U,f)$ and $(V,g)$ are representations of $(\Gamma, \Lambda)$ the direct sum of $(U,f) \oplus (V,g)$ is the representation $(X,s)$ where $X_\alpha = U_\alpha \oplus V_\alpha, \, \alpha \in \Gamma_0$ and $s_\ell: X_{\alpha(\ell)} \to X_{\beta(\ell)}$ is the linear map $s_\ell = f_\ell \oplus g_\ell: U_{\alpha(\ell)} \oplus V_{\alpha(\ell)} \to U_{\beta(\ell)} \oplus V_{\beta_\ell}$ where $\ell \in \Gamma_1$. Since the direct sums exist $\mathscr L(\Gamma,\Lambda)$ is an additive $k$-category. An object is \textbf{indecomposable} if it is not isomorphic to the direct sum of two nonzero representations.

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\section{Reflection Functors}

We present the following facts from the "Coxeter Functors and Gabriel's Theorem" paper written by Bernstein, Gel'fand, and Ponomarev. \\

For each vertex $\alpha \in \Gamma_0$ we denote by $\Gamma^\alpha$ the set of edges containing $\alpha$. If $\Lambda$ is some orientation of the graph $\Gamma$, we denote by $\sigma_\alpha\Lambda$ the orientation obtained from $\Lambda$ by changing the directions of all edges $\ell \in \Gamma^\alpha$.

\par We say that a vertex $\alpha$ is a source of $(\Gamma, \Lambda)$ if $\beta(\ell) \neq \alpha$ for all $\ell \in \Gamma_1$ (this means that all the edges containing $\alpha$ start there and that there are no loops in $\Gamma$ with vertex at $\alpha$). Similarly we say that a vertex $\beta$ is a sink of $(\Gamma, \Lambda)$ if $\alpha(\ell) \neq \beta$, for all $\ell \in \Gamma_1$.

\par To study indecomposable objects in the category $\mathscr L$$(\Gamma,\Lambda) we consider \textbf{refection functors} F^+_\beta :$$\mathscr L$$(\Gamma,\Lambda)$$ \to $$\mathscr L$$(\Gamma,\sigma_\beta \Lambda)$ and $F^-_\alpha : $$\mathscr L$$(\Gamma,\Lambda)$$\to$$\mathscr L$$(\Gamma,\sigma_\alpha \Lambda). These functors send an indecomposible representation to either an indecomposible representation or to zero. We construct such a functor for each vertex \alpha at which all the edges have the same direction. We will prove that F_\beta^+ is a functor in section 3.1, and that F_\alpha^- is a functor in section 3.2. %------------------------------------------------------------------------------------------------------- \subsection{A Positive Reflection Functor} Suppose that the vertex \beta of the graph \Gamma is a sink with respect to the orientation \Lambda. From an object (U,f) in \mathscr L$$(\Gamma,\Lambda)$ we construct a new object $F_\beta^+(U,f) = (X,r)$ in $\mathscr L$$(\Gamma,\sigma_\beta\Lambda). \par Namely, we put X_\gamma = U_\gamma for \gamma \neq \beta. To construct X_\beta we consider all the edges \ell_1, \ell_2, \ldots , \ell_k that end at \beta (that is, all edges of \Gamma^\beta). We denote by X_\beta the subspace in the direct sum \underset{i = 1}{\overset{k}{\oplus}} U_{\alpha(\ell_i)} consisting of the vectors u = (u_1, \ldots, u_k) (here u_i \in U_{\alpha(\ell_i)}) for which f_{\ell_i}(u_1) + \ldots + f_{\ell_k}(u_k) = 0. In other words, if we denote by h_U the mapping h_U: \underset{i = 1}{\overset{k}{\oplus}} U_{\alpha(\ell_i)} \to U_\beta defined by the formula h_U (u_1, u_2, \ldots, u_k) = f_{\ell_1}(u_1) + \ldots + f_{\ell_k}(u_k), then X_\beta = Ker\,h_U. \par We now define the mappings r_{\ell_j}. For \ell_j \notin \Gamma^\beta we put r_{\ell_j} = f_{\ell_j}. If \ell = \ell_j \in \Gamma^\beta, then r_{\ell_j} is defined as the composition of the natural embedding \kappa_U: X_\beta \to \oplus U_{\alpha(\ell_i)} of X_\beta in \oplus U_{\alpha(\ell_i)} and the projection \pi_{U,{\alpha(\ell_j)}}: \oplus U_{\alpha(\ell_i)} \to U_{\alpha(\ell_j)} of the sum \oplus U_{\alpha(\ell_i)} onto the term U_{\alpha(\ell_j)} = X_{\alpha(\ell_j)}. In other words, r_{\ell_j} = \pi_{U,{\alpha(\ell_j)}} \kappa_U . We note that on all edges \ell_j \in \Gamma^\beta the orientation has been changed, that is, the resulting object (X,r) belongs to \mathscr L$$(\Gamma,\sigma_\beta\Lambda)$.
Let $\phi = (\phi_\alpha): (U,f) \to (V,g)$ be a morphism in $\mathscr L$$(\Gamma,\Lambda), let (X,r) = F^+_\beta(U,f) and (Y,s) = F^+_\beta (V,g). We construct F^+_\beta(\phi) = \xi = (\xi_\alpha)_{\alpha \in \Gamma_0}: (X,r) \to (Y,s). If \alpha \neq \beta, then X_\alpha = U_\alpha, Y_\alpha = V_\alpha, and we set \xi_\alpha = \phi_\alpha: U_\alpha \to V_\alpha. To construct \xi_\beta: X_\beta \to Y_\beta, we consider the following diagram of vector spaces and linear maps \begin{equation} \xymatrix{ X_\beta \ar[r]^{\kappa_U} \ar[d]_{\xi_\beta} & \oplus^k_{i = 1}U_{\alpha(\ell_i)} \ar[r]^{h_U} \ar[d]^{\oplus\phi_{\alpha(\ell_i)}} & U_\beta \ar[d]^{\phi_\beta} \\ Y_\beta \ar[r]^{\kappa_V} & \oplus^k_{i = 1}V_{\alpha(\ell_i)} \ar[r]^{h_V} & V_\beta } \end{equation} \noindent where X_\beta = Ker\,h_U, Y_\beta = Ker\,h_V, and \kappa_U and \kappa_V are the inclusion maps. It is easy to verify that the right square of the diagram commutes. \begin{center} \phi_\beta h_U = h_V(\oplus^k_{i=1}\phi_{\alpha(\ell_i)}) \end{center} \newpage \cfoot{} \lfoot{\thepage} \rfoot{} \noindent Since h_V( \underset{i = 1}{\overset{k}{\oplus}} \phi_{\alpha(\ell_i)})\kappa_U = \phi_\beta h_U\kappa_U = \phi_\beta0 = 0, the universal property of the kernel (see Introduction) says that there exists a unique k-linear map \xi_\beta: X_\beta \to Y_\beta satisfying \kappa_V\xi_\beta = (\underset{i = 1}{\overset{k}{\oplus}} \phi_{\alpha(\ell_i)})\kappa_U. This finishes the construction of \xi = F^+_\beta(\phi). We now verify that it is a morphism in \mathscr L$$(\Gamma,\sigma_\beta\Lambda)$. For each edge $\ell = \ell_j: \beta \to \alpha_{(\ell_j)}$ in $\Gamma^\beta$ (in the orientation $\sigma_\beta \Lambda$), we have

\begin{center}
$\xi_{\alpha(\ell_j)} \pi_{U,{\alpha(\ell_j)}}(u_1, \dots, u_k) = \xi_{\alpha(\ell_j)}(u_j) = \phi_{\alpha(\ell_j)}(u_j)$ and \\
$\pi_{V_{\alpha(\ell_j)}}[\oplus \phi_{\alpha(\ell_i)}](u_1, \dots, u_k) = \pi_{V_{\alpha(\ell_j)}}(\phi_{\alpha(\ell_1)}(u_1), \dots, \phi_{\alpha(\ell_k)}(u_k)) = \phi_{\alpha(\ell_j)}(u_j)$. Hence \\
$\xi_{\alpha(\ell_j)} \pi_{U,\alpha(\ell_j)} = \pi_{V,\alpha(\ell_j)} [\oplus \phi_{\alpha(\ell_i)}]$ and we have \\
$\xi_{\alpha(\ell_j)} r_{\ell_j} =\xi_{\alpha(\ell_j)} \pi_{U,{\alpha(\ell_j)}}\kappa_U = \pi_{V,\alpha(\ell_j)}[\oplus \phi_{\alpha(\ell_i)}]\kappa_U = \pi_{V,\alpha(\ell_j)} \kappa_V \xi_\beta = s_{\ell_j} \xi_\beta$.
\end{center}

For each edge $\ell \in \Gamma_1$ not incident to $\beta$, we have $\alpha(\ell) \neq \beta$, $\beta(\ell) \neq \beta$, so

\centerline{
\xymatrix{
U_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]_{\phi_{\alpha(\ell)}} & U_{\beta(\ell)} \ar[d]^{\phi_{\alpha(\ell)}} \\
V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)}
}
}

\noindent is a commutative diagram because $\phi: (U,f) \to (V,g)$ is a morphism. Hence the above construction yields the commutative diagram

\centerline{
\xymatrix{
X_{\alpha(\ell)} \ar[r]^{r_\ell} \ar[d]_{\xi_{\alpha(\ell)}} & X_{\beta(\ell)} \ar[d]^{\xi_{\beta(\ell)}} \\
Y_{\beta(\ell)} \ar[r]^{s_\ell} & Y_{\beta(\ell)}
}
}

\noindent as required.

We show that $F_\beta^+: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\beta\Lambda)$ satisfies the following conditions and therefore is a functor:

\begin{enumerate}
\item $F_\beta^+ (1_{(U,f)}) = 1_{(X,r)}$

\item $F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi))$

\end{enumerate}

As previously defined, $1_{(U,f)}: (U,f) \to (U,f)$, and $F^+_\beta(1_{(U,f)}) = \xi = (\xi_\alpha)_{\alpha \in \Gamma_0} : (X,r) \to (X,r)$. To show: $\xi_\alpha = 1_{X_\alpha}$, $\alpha \in \Gamma_0$.

If $\alpha \neq \beta$, then $\xi_\alpha = \phi_\alpha$, but $\phi_\alpha = 1_{U_\alpha} = 1_{X_\alpha}$ since $\alpha \neq \beta$.

To show $\xi_\beta = 1_{X_\beta}$, we specialize the diagram $(2)$ to the case where $\phi = 1_{(U,f)} : (U,f) \to (U,f)$. We obtain the following commutative diagram

\centerline{
\xymatrix{
X_\beta \ar[d]_{\xi_\beta} \ar[r]^{\kappa_U} & \oplus U_{\alpha(\ell_i)}  \ar[d]_{\oplus 1_{U_{\alpha(\ell_i)}}} \ar[r]^{h_U} & U_\beta \ar[d]_{1_{U_\beta}} \\
X_\beta \ar[r]^{\kappa_U} & \oplus U_{\alpha(\ell_i)} \ar[r]^{h_U} & U_\beta
}
}

\noindent It is clear that replacing $\xi_\beta$ with $1_{X_\beta}$ preserves the commutativity of the left square of the diagram: $\kappa_U 1_{X_\beta} = (\oplus 1_{U_{\alpha(\ell_i)}}) \kappa_U = (1_{\oplus U_{\alpha(\ell_i)}}) \kappa_U = \kappa_U$. By the uniqueness of $\xi_\beta$ we must have $\xi_\beta = 1_{X_\beta}$.

\noindent Hence, $F^+_\beta(1_{(U,f)}) = 1_{(X,r)}$. \\[12pt]

\newpage

\cfoot{}
\rfoot{\thepage}
\lfoot{}

\noindent Now we check if $F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi))$. \\
\noindent For any objects $(U,f)$, $(V,g)$, and $(W,h)$ in $\mathscr L$$(\Gamma,\Lambda), let \phi: (U,f) \to (V,g) and \\ \psi: (V,g) \to (W,h) be morphisms. \noindent Set \begin{center} F_\beta^+(\phi) = \xi = (\xi_\alpha)_{\alpha \in \Gamma_0} \\ F_\beta^+(\psi) = \zeta = (\zeta_\alpha)_{\alpha \in \Gamma_0} \\ F_\beta^+(\psi \phi) = \theta = (\theta_\alpha)_{\alpha \in \Gamma_0} \end{center} \noindent We want to show that \theta_\alpha = \zeta_\alpha \xi_\alpha, \alpha \in \Gamma_0. \noindent a) For \alpha \neq \beta \begin{center} \theta_\alpha = [F_\beta^+(\psi \phi)]_\alpha = (\psi \phi)_\alpha = \psi_\alpha \phi_\alpha = [F_\beta^+(\psi)]_\alpha [F_\beta^+(\phi)]_\alpha = \zeta_\alpha \xi_\alpha \end{center} \noindent b) For \alpha = \beta we set X_\beta = Ker\,h_U, Y_\beta = Ker\,h_V, and Z_\beta = Ker\,h_W \centerline{ \xymatrix{ X_\beta \ar[d]_{\xi_\beta} \ar[r]^{\kappa_U} & \oplus^k_{i=1} U_{\alpha(\ell_i)} \ar[d]_{\oplus^k_{i=1} \phi_{\alpha(\ell_i)}} \ar[r]^{h_U} & U_\beta \ar[d]^{\phi_\beta} \\ Y_\beta \ar[d]_{\zeta_\beta} \ar[r]^{\kappa_V} & \oplus^k_{i=1} V_{\alpha(\ell_i)} \ar[d]_{\oplus^k_{i=1} \psi_{\alpha(\ell_i)}} \ar[r]^{h_V} & V_\beta \ar[d]^{\psi_\beta} \\ Z_\beta \ar[r]^{\kappa_W} & \oplus^k_{i=1} W_{\alpha(\ell_i)} \ar[r]^{h_W} & W_\beta } } \noindent By (2) the above diagram commutes so \begin{center} [\oplus(\psi\phi)_{\alpha(\ell_i)}] \kappa_U = (\underset{i = 1}{\overset{k}{\oplus}} \psi_{\alpha(\ell_i)}\phi_{\alpha(\ell_i)}) \kappa_U = (\underset{i = 1}{\overset{k}{\oplus}} \psi_{\alpha(\ell_i)}) (\underset{i = 1}{\overset{k}{\oplus}} \phi_{\alpha(\ell_i)}) \kappa_U = (\underset{i = 1}{\overset{k}{\oplus}} \psi_{\alpha(\ell_i)}) \kappa_V \xi_\beta = \kappa_W \zeta_\beta \xi_\beta \end{center} By (2), the diagram below commutes. \\ \centerline{ \xymatrix{ X_\beta \ar[d]_{\theta_\beta} \ar[r]^{\kappa_U} & \underset{i = 1}{\overset{k}{\oplus}} U_{\alpha(\ell_i)} \ar[d]_{\underset{i = 1}{\overset{k}{\oplus}} (\psi\phi)_{\alpha(\ell_i)}} \ar[r]^{h_U} & U_\beta \ar[d]^{(\psi\phi)_\beta} \\ Z_\beta \ar[r]^{\kappa_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\alpha(\ell_i)} \ar[r]^{h_W} & W_\beta } } \noindent We have \begin{center} [\oplus (\psi \phi)_{\alpha(\ell_i)}] \kappa_U = \kappa_W \theta_\beta \end{center} So both \zeta_\beta \xi_\beta and \theta_\beta make the left square of the above diagram commute. By the uniqueness of \theta_\beta, we must have \theta_\beta = \zeta_\beta \xi_\beta. Therefore F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi)) and F_\beta^+ (1_{(U,f)}) = 1_{(X,r)}. Thus F_\beta^+ is a functor. It is easy to see that F_\beta^+(\phi + \psi) = F_\beta^+(\phi) + F_\beta^+(\psi) and F_\beta^+(c\phi) = cF_\beta^+(\phi). Therefore F_\beta^+ is a k-linear functor. %% ---------------------------------------------------------------------------------------------- \newpage \cfoot{} \lfoot{\thepage} \rfoot{} \subsection{A Negative Reflection Functor} Suppose that the vertex \alpha of the graph \Gamma is a source with respect to the orientation \Lambda. From an object (U,f) in \mathscr L$$(\Gamma,\Lambda)$ we construct a new object $F^-_\alpha (U,f) = (X,r)$ in $\mathscr L$$(\Gamma,\sigma_\alpha\Lambda). \par Namely, we put X_\gamma = U_\gamma for \gamma \neq \alpha. \par Next we consider all the edges \ell_1, \ell_2, \ldots , \ell_k that start at \alpha (that is, all edges of \Gamma^\alpha). We denote by \widetilde{h}_U the mapping \widetilde{h}_U : U_\alpha \to \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} defined by the formula \widetilde{h}_U(u) = (f_{\ell_1}(u), \ldots, f_{\ell_k}(u)), and set X_\alpha = Coker\, \widetilde{h}_U = \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)}/Im\,\widetilde{h}_U. Denote by \pi_U : \oplus U_{\beta(\ell_i)} \to X_\alpha the canonical map. \par We now define the mappings r_\ell. For \ell \notin \Gamma^\alpha we put r_\ell = f_\ell. If \ell = \ell_j \in \Gamma^\alpha, then r_{\ell_j} is defined as the composition of the natural embedding \kappa_{U, \ell_j} : U_{\beta(\ell_j)} \to \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} and the canonical map \pi_{U}: \oplus U_{\beta(\ell_i)} \to X_\alpha. In other words, r_{\ell_j} = \pi_U \kappa_{U,\beta(\ell_j)}. We note that on all edges \ell \in \Gamma^\alpha the orientation has been changed, that is, the resulting object (X,r) belongs to \mathscr L$$(\Gamma,\sigma_\alpha\Lambda)$. Let $\phi = (\phi_\beta): (U,f) \to (V,g)$ be a morphism in $\mathscr L$$(\Gamma,\Lambda), let (X,r) = F^-_\alpha(U,f) and (Y,s) = F^-_\alpha (V,g). We construct F^-_\alpha(\phi) = \xi = (\xi_\beta)_{\beta \in \Gamma_0}: (X,r) \to (Y,s). If \beta \neq \alpha, then X_\beta = U_\beta, Y_\beta = V_\beta, and we set \xi_\beta = \phi_\beta: U_\beta \to V_\beta. To construct \xi_\alpha: X_\alpha \to Y_\alpha, we consider the following diagram of vector spaces and linear maps \begin{equation} \xymatrix{ U_\alpha \ar[r]^{\widetilde{h_U}} \ar[d]_{\phi_\alpha} & \oplus^k_{i = 1}U_{\beta(\ell_i)} \ar[r]^{\pi_{U}} \ar[d]^{\oplus\phi_{\beta(\ell_i)}} & X_\alpha \ar[d]^{\xi_\alpha} \\ V_\alpha \ar[r]^{\widetilde{h_V}} & \oplus^k_{i = 1}V_{\beta(\ell_i)} \ar[r]^{\pi_{V}} & Y_\alpha } \end{equation} \noindent where X_\alpha = Coker\,\widetilde{h_U}, Y_\alpha = Coker\,\widetilde{h_V}, and \pi_{U} and \pi_{U} are the canonical maps. It is easy to verify that the left square of the diagram commutes. \begin{center} \widetilde{h_V} \phi_\alpha = (\oplus^k_{i=1}\phi_{\beta(\ell_i)}) \widetilde{h_U} \end{center} \noindent Since \pi_V (\oplus \phi_{\beta(\ell_i)}) \widetilde{h}_U = \pi_V \widetilde{h}_V \phi_\alpha = 0, the universal property of the cokernel (see Introduction) says that there exists a unique k-linear map \xi_\alpha: X_\alpha \to Y_\alpha satisfying \pi_V(\oplus \phi_{\beta(\ell_i)}) = \xi_\alpha \pi_U. This finishes the construction of \xi = F^-_\alpha(\phi). We now verify that it is a morphism in \mathscr L$$(\Gamma,\sigma_\alpha\Lambda)$. For each edge $\ell = \ell_j: \beta_{(\ell_j)} \to \alpha$ in $\Gamma^\alpha$ (in the orientation $\sigma_\alpha \Lambda$), we claim that

$$[\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_j)} = \kappa_{V, \beta(\ell_j)} \xi_{\beta(\ell_j)}.$$

\noindent Indeed

$$[\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_i)}(u_j) = [\oplus \phi_{\beta(\ell_i)}](0, \ldots, u_j, \ldots, 0) = (0, \ldots, \phi_{\beta(\ell_j)}(u_j), \ldots, 0)$$

\noindent and

$$\kappa_{V, \beta(\ell_j)} \xi_{\beta(\ell_i)}(u_j) = \kappa_{V, \beta(\ell_j)} (\phi_{\beta(\ell_i)} (u_j)) = (0, \ldots , \phi_{\beta(\ell_j)}(u_j), \ldots, 0).$$

\noindent Therefore

\begin{center}
$\xi_\alpha r_{\ell_j} = \pi_V [\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_j)} = \pi_V \kappa_{V,\beta(\ell_j)} \xi_{\beta(\ell_j)} = s_{\ell_j} \xi_{\beta(\ell_j)}$.

\end{center}

\newpage

\cfoot{}
\rfoot{\thepage}
\lfoot{}

For each edge $\ell \in \Gamma_1$ not incident to $\alpha$, we have $\beta(\ell) \neq \alpha$, $\alpha(\ell) \neq \alpha$, so

\centerline{
\xymatrix{
U_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]_{\phi_{\alpha(\ell)}} & U_{\beta(\ell)} \ar[d]^{\phi_{\alpha(\ell)}} \\
V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)}
}
}

\noindent is a commutative diagram because $\phi: (U,f) \to (V,g)$ is a morphism. Hence the above construction yield the commutative diagram

\centerline{
\xymatrix{
X_{\alpha(\ell)} \ar[r]^{r_\ell} \ar[d]_{\xi_{\alpha(\ell)}} & X_{\beta(\ell)} \ar[d]^{\xi_{\beta(\ell)}} \\
Y_{\beta(\ell)} \ar[r]^{s_\ell} & Y_{\beta(\ell)}
}
}

\noindent as required.

We show that $F_\alpha^-: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\alpha\Lambda)$ satisfies the following conditions and therefore is a functor:

\begin{enumerate}
\item $F_\alpha^- (1_{(U,f)}) = 1_{(X,r)}$

\item $F_\alpha^- (\psi\phi) = (F_\alpha^- (\psi))(F_\alpha^-(\phi))$

\end{enumerate}

As previously defined, $1_{(U,f)}: (U,f) \to (U,f)$, and $F^-_\alpha(1_{(U,f)}) = \xi = (\xi_\beta)_{\beta \in \Gamma_0} : (X,r) \to (X,r)$. To show: $\xi_\beta = 1_{X_\beta}$, $\beta \in \Gamma_0$.

If $\beta \neq \alpha$, then $\xi_\beta = \phi_\beta$, but $\phi_\beta = 1_{U_\beta} = 1_{X_\beta}$ since $\beta \neq \alpha$.

To show $\xi_\alpha = 1_{X_\alpha}$, we specialize the diagram $(3)$ to the case where $\phi = 1_{(U,f)} : (U,f) \to (U,f)$. We obtain the following commutative diagram

\centerline{
\xymatrix{
U_\alpha \ar[d]_{1_{U_\alpha}} \ar[r]^{\widetilde{h}_U} & \oplus U_{\beta(\ell_i)}  \ar[d]_{\oplus 1_{U_{\beta(\ell_i)}}} \ar[r]^{\pi_U} & X_\alpha \ar[d]_{\xi_\alpha} \\
U_\alpha \ar[r]^{\widetilde{h}_U} & \oplus U_{\beta(\ell_i)} \ar[r]^{\pi_U} & X_\alpha
}
}

\noindent It is clear that replacing $\xi_\alpha$ with $1_{X_\alpha}$ preserves the commutativity of the right square of the diagram: $\pi_U = 1_{X_\alpha} \pi_U = \pi_U (1_{\oplus U_{\beta(\ell_i)}}) = \pi_U (\oplus 1_{U_{\beta(\ell_i)}})$. By the uniqueness of $\xi_\alpha$ we must have $\xi_\alpha = 1_{X_\alpha}$.

\noindent Hence, $F^-_\alpha(1_{(U,f)}) = 1_{(X,r)}$. \\[12pt]

\noindent Now we check if $F_\alpha^- (\psi\phi) = (F_\alpha^- (\psi))(F_\alpha^-(\phi))$. \\
\noindent For any objects $(U,f)$, $(V,g)$, and $(W,h)$ in $\mathscr L$$(\Gamma,\Lambda), let \phi: (U,f) \to (V,g) and \\ \psi: (V,g) \to (W,h) be morphisms. \noindent Set \begin{center} F_\alpha^-(\phi) = \xi = (\xi_\beta)_{\beta \in \Gamma_0} \\ F_\alpha^-(\psi) = \zeta = (\zeta_\beta)_{\beta \in \Gamma_0} \\ F_\alpha^-(\psi \phi) = \theta = (\theta_\beta)_{\beta \in \Gamma_0} \end{center} \noindent We want to show that \theta_\beta = \zeta_\beta \xi_\beta, \beta \in \Gamma_0. \noindent a) For \beta \neq \alpha \newpage \cfoot{} \lfoot{\thepage} \rfoot{} \begin{center} \theta_\beta = [F_\alpha^-(\psi \phi)]_\beta = (\psi \phi)_\beta = \psi_\beta \phi_\beta = [F_\alpha^-(\psi)]_\beta [F_\alpha^-(\phi)]_\beta = \zeta_\beta \xi_\beta \end{center} \noindent b) For \beta = \alpha we set X_\alpha = Coker\, \widetilde{h}_U, Y_\alpha = Coker\, \widetilde{h}_V, and Z_\alpha = Coker\, \widetilde{h}_W \centerline{ \xymatrix{ U_\alpha \ar[d]_{\phi_\alpha} \ar[r]^{\widetilde{h}_U} & \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} \ar[d]_{\underset{i = 1}{\overset{k}{\oplus}} \phi_{\beta(\ell_i)}} \ar[r]^{\pi_U} & X_\alpha \ar[d]^{\xi_\alpha} \\ V_\alpha \ar[d]_{\psi_\alpha} \ar[r]^{\widetilde{h}_V} & \underset{i = 1}{\overset{k}{\oplus}} V_{\beta(\ell_i)} \ar[d]_{\underset{i = 1}{\overset{k}{\oplus}} \psi_{\beta(\ell_i)}} \ar[r]^{\pi_V} & Y_\alpha \ar[d]^{\zeta_\alpha} \\ W_\alpha \ar[r]^{\widetilde{h}_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\beta(\ell_i)} \ar[r]^{\pi_W} & Z_\alpha } } \noindent By (3) the above diagram commutes so \begin{center} \pi_W [\oplus(\psi\phi)_{\beta(\ell_i)}] = \pi_W (\underset{i = 1}{\overset{k}{\oplus}} \psi_{\beta(\ell_i)}\phi_{\beta(\ell_i)}) = \pi_W (\underset{i = 1}{\overset{k}{\oplus}} \psi_{\beta(\ell_i)}) (\underset{i = 1}{\overset{k}{\oplus}} \phi_{\beta(\ell_i)}) = \zeta_\alpha \pi_V (\underset{i = 1}{\overset{k}{\oplus}} \phi_{\beta(\ell_i)}) = \zeta_\alpha \xi_\alpha \pi_U \end{center} By (3), the diagram below commutes. \\ \centerline{ \xymatrix{ U_\alpha \ar[d]_{(\psi \phi)_\alpha} \ar[r]^{\widetilde{h}_U} & \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} \ar[d]_{\underset{i = 1}{\overset{k}{\oplus}} (\psi\phi)_{\beta(\ell_i)}} \ar[r]^{\pi_U} & X_\alpha \ar[d]^{\theta_\alpha} \\ W_\alpha \ar[r]^{\widetilde{h}_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\beta(\ell_i)} \ar[r]^{\pi_W} & Z_\alpha } } \noindent We have \begin{center} \pi_W [\oplus (\psi \phi)_{\beta(\ell_i)}] = \theta_\alpha \pi_U \end{center} So both \zeta_\alpha \xi_\alpha and \theta_\alpha make the left square of the above diagram commute. By the uniqueness of \theta_\alpha, we must have \theta_\alpha = \zeta_\alpha \xi_\alpha. Therefore F_\alpha^- (\psi\phi) = (F_\alpha^- (\psi))(F_\alpha^-(\phi)) and F_\alpha^- (1_{(U,f)}) = 1_{(X,r)}. Thus F_\alpha^- is a functor. It is easy to see that F_\alpha^-(\phi + \psi) = F_\alpha^-(\phi) + F_\alpha^-(\psi) and F_\alpha^-(c\phi) = cF_\alpha^-(\phi). Therefore F_\alpha^- is a k-linear functor. %----------------------------------------------------------------------------------------------- \subsection{Properties of Reflection Functors} Let (\Gamma, \Lambda) be a quiver. For each \gamma \in \Gamma_0 we denote by L_\gamma a simple representation defined by the condition (L_\gamma)_\delta = 0 for \delta \neq \gamma, (L_\gamma)_\gamma = K, f_\ell = 0 for all \ell \in \Gamma_1. \begin{theorem} 1) Let (\Gamma, \Lambda) be a quiver and let \beta \in \Gamma_0 be a sink. Let V \in \mathscr L$$(\Gamma,\Lambda)$ be an indecomposable representation. Then two cases are possible: \vskip.05in
\noindent $a) \, V \approx L_\beta$ and $F^+_\beta V = 0$. \vskip.05in
\noindent $b) \, F^+_\beta (V)$ is an indecomposable representation, $F^-_\beta F^+_\beta (V) = V$, and the dimensions of the spaces $F^+_\beta (V)_\gamma$ can be calculated by the formula

\newpage
\cfoot{}
\rfoot{\thepage}
\lfoot{}

\begin{center}

$\dim F^+_\beta(V)_\gamma = \dim V_\gamma \, \text{ for} \, \gamma \neq \beta$, \\
$\dim F^+_\beta(V)_\beta = -\dim V_\beta + \underset{\ell \in \Gamma^\beta}{\Sigma} \dim V_{\alpha(\ell)}$.

\end{center}

$2)$ If the vertex $\alpha$ is a source, and if $V \in $$\mathscr L$$(\Gamma,\Lambda)$ is an indecomposable representation, then two cases are possible: \vskip.05in
\noindent $a) \, V \approx L_\alpha$ and $F^-_\alpha (V) = 0$. \vskip.05in
\noindent $b) \, F^-_\alpha (V)$ is an indecomposable representation, $F^+_\alpha F^-_\alpha (V) = V$, and
\begin{center}

$\dim F^-_\alpha (V)_\gamma = \dim V_\gamma$ for $\gamma \neq \alpha$, \\
$\dim F^-_\alpha (V)_\alpha = -\dim V_\alpha + \underset{\ell \in \Gamma^\alpha}{\Sigma}$ $\dim V_{\beta(\ell)}$.

\end{center}

\end{theorem}

Proof. If the vertex $\beta$ is a sink with respect to $\Lambda$, then it is a source with respect to $\sigma_\beta \Lambda$, and so the functor $F_\beta^- F_\beta^+$: $\mathscr L$$(\Gamma,\Lambda) \to \mathscr L$$(\Gamma,\Lambda)$ is defined. For each representation $(V, g) \in$ $\mathscr L$$(\Gamma,\Lambda) we set (Y, s) = F^+_\beta (V, g) and (Z, t) = F^-_\beta (Y, s) so that Z_\beta = (F_\beta^-(Y))_\beta = (F^-_\beta(F^+_\beta(V)))_\beta = (F_\beta^- F_\beta^+)(V)_\beta. We construct a morphism i^\beta_V: F_\beta^- F_\beta^+ (V, g) \to (V, g). If \gamma \neq \beta, then F_\beta^- F_\beta^+ (V)_\gamma = V_\gamma, and we put (i_V^\beta)_\gamma = Id, the identity mapping. For the definition of (i_V^\beta)_\beta, we consider the following diagram of K-vector spaces. \begin{equation} \label{4} \xymatrix{ Y_\beta \ar[rr]^{\widetilde{h}_Y = \kappa_V} & & \underset{i = 1}{\overset{k}{\oplus}} V_{\alpha(\ell_i)} \ar[rr]^{\pi_Y} \ar[rrd]^{h_V} & & Z_\beta \[email protected]{-->}[d]^{(i_V^\beta)_\beta} \\ & & & & V_\beta } \end{equation} Here the notation is the same as that of formulas (2) and (3). In particular, Y_\beta = Ker\, h_V and Z_\beta = Coker\, \widetilde{h}_Y = \underset{i = 1}{\overset{k}{\oplus}} V_{\alpha(\ell_i)}/Ker h_V. By the First Isomorphism Theorem, there exists a unique linear map (i_V^\beta)_\beta satisfying h_V = (i_V^\beta)_\beta \pi_Y. Now we check that i_V^\beta is a morphism. Let \ell \in \Gamma_1, we want to show that the diagram \centerline{ \xymatrix{ Z_{\alpha(\ell)} \ar[d]_{(i_V^\beta)_{\alpha(\ell)}} \ar[r]^{t_\ell} & Z_{\beta(\ell)} \ar[d]^{(i_V^\beta)_{\beta(\ell)}} \\ V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)} } } \noindent commutes. If \ell \notin \Gamma^\beta, the verification is trivial, and we leave it to the reader. Let now \ell = \ell_j \in \Gamma^\beta. Then \alpha(\ell_j) \neq \beta so that Z_{\alpha(\ell_j)} = V_{\alpha(\ell_j)} and (i^\beta_V)_{\alpha(\ell_j)} = Id. Since V_{\alpha(\ell_i)} = Y_{\alpha(\ell_i)} for all i, the formulas preceding diagram (3) say that t_{\ell_j} = \pi_Y \kappa_{V, \alpha(\ell_j)}. Then \begin{center} (i_V^\beta)_{\beta} t_{\ell_j} = (i_V^\beta)_{\beta} \pi_Y \kappa_{V, \alpha(\ell_j)} = h_V \kappa_{V, \alpha(\ell_j)} \end{center} \noindent The latter equality holds, for if v \in V_{\alpha(\ell_j)}, then h_V \kappa_{V, \alpha(\ell_j)} (v) = h_V (0, \ldots , v, \ldots , 0) = g_{\ell_1} (0) + \ldots + g_{\ell_j} (v) + \ldots + g_{\ell_k} (0) = g_{\ell_j} (v). Here we used the formulas preceding the diagram (2). Therefore the diagram below commutes. \centerline{ \xymatrix{ V_{\alpha(\ell)} \ar[r]^{\pi_Y \kappa_{V, \alpha(\ell)}} \ar[d]_{Id} & Z_\beta \ar[d]^{(i_V^\beta)_\beta} \\ V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_\beta } } \newpage \cfoot{} \lfoot{\thepage} \rfoot{} Similarly, for each source vertex \alpha we construct a morphism p_V^\alpha: V \to F_\beta^- F_\beta^+ (V). Now we state the basic properties of the functors F_\alpha^-, F_\beta^+ and the morphisms p_V^\alpha, i^\beta_V. \begin{lemma} 1) F^\pm_\alpha (V_1 \oplus V_2) = F^\pm_\alpha(V_1) \oplus F^\pm_\alpha(V_2). \\ 2) p_V^\alpha is an epimorphism and i^\beta_V is a monomorphism. \\ 3) If i_V^\beta is an isomorphism, then the dimensions of the spaces F_\beta^+(V)_\gamma can be calculated from (1.1.1). If p_V^\alpha is an isomorphism, then the dimensions of the spaces F_\alpha^- (V)_\gamma can be calculated from (1.1.2). \\ 4) The object Ker\, p_V^\alpha is concentrated at \alpha (that is, (Ker\, p_V^\alpha)_\gamma = 0 for \gamma \neq \alpha). The representation V/Im\, i_V^\beta is concentrated at \beta. \\ 5) If the representation V has the form F_\beta^- W (F^+_\alpha W respectively), then i^\beta_V (p^\alpha_V) is an isomorphism. \\ 6) The representation V is isomorphic to the direct sum of the representations F^-_\beta F^+_\beta (V) and V/Im\, i_V^\beta (similarly, V \approx F^+_\alpha F^-_\alpha (V) \oplusKer\, p_V^\alpha). \end{lemma} Say how define direct sum in category, then use fact about categories then use that they are additive functors to prove 1. For 2 we have that for it to be an whatever all of it's parts also have to be an whatever. Proof. 1) We recall the direct sum construction for quiver representations. If V_1 = (V_1, g_1), V_2 = (V_2, g_2), we define V_1 \oplus V_2 = (V_1 \oplus V_2, h) as follows. For all \gamma \in \Gamma_0, (V_1 \oplus V_2)_\gamma = (V_1)_\gamma \oplus (V_2)_\gamma, and for all \ell \in \Gamma_1, \ell: \alpha(\ell) \to \beta(\ell), h_\ell = (g_1)_\ell \oplus (g_2)_\ell : (V_1)_{\alpha(\ell)} \oplus (V_2)_{\alpha(\ell)} \to (V_1)_{\beta(\ell)} \oplus (V_2)_{\beta(\ell)}. The maps \iota_1: (V_1, g_1) \to (V_1 \oplus V_2, h) and \pi_1: (V_1 \oplus V_2, h) \to (V_1, g_1) are defined as follows. For each \gamma \in \Gamma_0, (i_1)_\gamma : (V_1)_\gamma \to (V_1 \oplus V_2)_\gamma = (V_1)_\gamma \oplus (V_2)_\gamma is given by (i_1)_\gamma (a) = (a, 0), and (\pi_1)_\gamma : (V_1)_\gamma \oplus (V_2)_\gamma \to (V_1)_\gamma is given by (\pi_1)_\gamma (a,b) = a. Then we define linear maps i_2 : (V_2, g_2) \to (V_1, g_1) \oplus (V_2, g_2) and \pi_2: (V_1, g_1) \oplus (V_2, g_2) \to (V_2, g_2) analogously. We leave it to the reader to verify that \iota_j, \pi_j, j = 1, 2, are morphisms in \mathscr L$$(\Gamma, \Lambda)$, and that $\pi_j \iota_j = 1_{(V_j, g_j)}, j = 1,2,$ as well as $\iota_1 \pi_1 + \iota_2 \pi_2 = 1_{(V_1 \oplus V_2, h)}$.

Since we know that $\mathscr L$$(\Gamma, \Lambda) and \mathscr L$$(\Gamma, \sigma_\alpha \Lambda)$ are additive categories and that $F^+_\alpha$ and $F^-_\alpha$ are additive functors then the statement is a consequence of the following general result.

\begin{proposition} Let $\mathscr B$ and $\mathscr C$ be preadditive categories, and $F: \mathscr B \to \mathscr C$ an additive functor. If $A_1 \underset{\pi_1}{\overset{\iota_1}{\rightleftarrows}} A \underset{\pi_2}{\overset{\iota_2}{\leftrightarrows}} A_2$ is a direct sum diagram in $\mathscr B$, then $FA_1 \underset{F\pi_1}{\overset{F\iota_1}{\rightleftarrows}} FA \underset{F\pi_2}{\overset{F\iota_2}{\leftrightarrows}} FA_2$ is a direct sum diagram in $\mathscr C$.
\end{proposition}
Proof: By assumption, $\pi_j \iota_j = 1_{A_j}$ for $j = 1,2$, and $\iota_1 \pi_1 + \iota_2 \pi_2 = 1_{A}$. Applying $F$, we get

\begin{center}
$F\pi_j Fi_j = F(\pi_ji_j) = F(1_{A_j}) = 1_{F A_j}$ \\
$Fi_1 F\pi_1 + Fi_2 F\pi_2 = F(i_1 \pi_1) + F(i_2 \pi_2) = F(i_1 \pi_1 + i_2 \pi_2) = F(1_A) = 1_{FA}$
\end{center}
\vskip.1in

$2)$ To show that $i_V^\beta$ is a monomorphism we need to check that all of its components are monomorphisms. Since $(i_V^\beta)_\gamma =$ Id for $\gamma \neq \beta$, clearly the identity is a monomorphism. The First Isomorphism Theorem says that the map $(i_V^\beta)_\beta$ in diagram $(4)$ is a monomorphism. Therefore $i_V^\beta$ is a monomorphism.
\par Similarly it is easy to verify that $p_V^\alpha$ is an epimorphism.
\vskip.1in

\newpage

\cfoot{}
\rfoot{\thepage}
\lfoot{}

$3)$ The first of the two formulas in Theorem $1$ part $1b)$ is obvious. Since $i_V^\beta$ is an isomorphism by assumption, then $(i_V^\beta)_\beta$ is an isomrophism of vector spaces. We know from diagram $(4)$ that $h_V = (i_V^\beta)_\beta \pi_Y$, where $\pi_Y$ and $(i_V^\beta)_\beta$ are both epimorphisms, so $h_V$ is a composition of epimorphisms making it an epimorhpism. Therefore we obtain an exact sequence of vector spaces.

\centerline{
\xymatrix{
0 \ar[r] & F_\beta^+(V)_\beta \ar[r] & \oplus_{i = 1}^{k} V_{\alpha(\ell_i)} \ar[r] & V_\beta \ar[r] & 0
}
}

\noindent Then $\dim \underset{i = 1}{\overset{k}{\oplus}} V_{\alpha(\ell_i)} = \underset{i = 1}{\overset{k}{\sum}} \dim V_{\alpha(\ell_i)} = \dim F_\beta^+(V)_\beta + \dim V_\beta$. The Theorem $1$ part $1b)$ equations follow.

\par Likewise, if $p_V^\alpha$ is an isomorphism then the equations from Theorem $1$ part $2b)$ holds.

\vskip.1in

$4)$ When $\gamma \neq \alpha$ then $(p_V^\alpha)_\gamma =$ Id, therefore $($Ker$\, p_V^\alpha)_\gamma = 0$. For each $\gamma \neq \beta$ we have $(i_V^\beta)_\gamma =$ Id$: V_\gamma \to V_\gamma$, therefore $($Im$\,i_V^\beta)_\gamma = V_\gamma$ so that $(V/$Im$\, i_V^\beta)_\gamma = V_\gamma/V_\gamma = 0$.

\vskip.1in

$5)$ When $\gamma \neq \beta$ then $(i_V^\beta)_\gamma =$ Id which is an isomorphism. Since $V_\beta$ is obtained by a negative reflection functor, the map $h_V$ in diagram $(4)$ is an epimorphism. Since $h_V = \pi_Y (i_V^\beta)_\beta$ then $(i_V^\beta)_\beta$ must be a epimorphism. Since we know $(i_V^\beta)_\beta$ is a monomorphism then $i_V^\beta$ is an isomorphism.

\par Similarly, the statement regarding $p_V^\alpha$ holds.

\vskip.1in

$6)$ We have to show that $V \approx F^+_\alpha F^-_\alpha (V) \oplus \widetilde{V}$, where $\widetilde{V} = V/$Im$\, i^\beta_V$. The natural projection $\phi'_\beta$: $V_\beta \to \widetilde{V}_\beta$ has a section $\phi_\beta$: $\widetilde{V}_\beta \to V_\beta$ ($\phi'_\beta \phi_\beta =$ Id). If we put $\phi_\gamma = 0$ for $\gamma \neq \beta$, we obtain a morphism $\phi: \widetilde{V} \to V$. It is clear that the morphisms $\phi: \widetilde{V} \to V$ and $i_V^\beta: F_\beta^- F_\beta^+ (V) \to V$ give a decomposition of $V$ into a direct sum. We can prove similarly that $V \approx F^+_\alpha F^-_\alpha (V) \oplus$ Ker$\, p_V^\alpha$.
We now prove Theorem $1$. Let $V$ be an indecomposable representation of the category $\mathscr L$$(\Gamma, \Lambda), and \beta a sink vertex with respect to \Lambda. Since V \approx F^-_\beta F^+_\beta (V) \oplus V/Im\, i_V^\beta and V is indecomposable, V coincides with one of the terms. \\ Case I). V = V/Im\, i_V^\beta. Then V_\gamma = 0 for \gamma \neq \beta and, because V is indecomposable, V \approx L_\beta. \\ Case II). V = F^-_\beta F^+_\beta (V), that is, i_V^\beta is an isomorphism. Then (Theorem 1 part 1) is satisfied by Lemma 1. We show that the representation W = F_\beta^+ (V) is indecomposable. For suppose that W = W_1 \oplus W_2. Then V = F^-_\beta (W_1) \oplus F^-_\beta (W_2) and so one of the terms (for example, F^-_\beta (W_2)) is 0. By (5) of Lemma 1 the morphism p_V^\beta : W \to F^+_\beta F^-_\beta (W) is an isomorphism, but p_V^\beta (W_2) \subset F^+_\beta F^-_\beta (W_2) = 0, that is, W_2 = 0. So we have shown that the representation F^+ _\beta (V) is indecomposable. We can similarly prove (2) of Theorem 1. \par We say that a sequence of vertices \alpha_1, \ldots, \alpha_k is a sink with respect to \Lambda if \alpha_1 is a sink with respect to \Lambda, \alpha_2 is a sink with respect to \sigma_{\alpha_1} \Lambda, \alpha_3 is a sink with respect to \sigma_{\alpha_2} \sigma_{\alpha_1} \Lambda, and so on. We define a source sequence similarly. \begin{corollary} Let (\Gamma, \Lambda) be an oriented graph and \alpha_1, \alpha_2, \ldots, \alpha_k a sink sequence. 1) For any i (1 \leq i \leq k), F^-_{\alpha_1} \cdot \ldots \cdot F^-_{\alpha_{i - 1}} (L_{\alpha_i}) is either 0 or an indecomposable representation in \mathscr L$$(\Gamma, \Lambda)$ (here $L_{\alpha_i} \in \mathscr L (\Gamma, \sigma_{\alpha_{i - 1}} \ldots \sigma_{\alpha_{1}} \Lambda$)). \\
$2)$ Let $V \in \mathscr L (\Gamma, \Lambda)$ be an indecomposable representation, and \\
\centerline{
$F^+_{\alpha_k} \cdot \ldots \cdot F^+_{\alpha_1} (V) = 0$
}
Then for some $i$ \\
\centerline{
$V \approx F^-_{\alpha_1} \cdot \ldots \cdot F^-_{\alpha_{i - 1}} (L_{\alpha_i})$.
}

\end{corollary}

%----------------------------------------------------------------------------------------------

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Let $\Gamma$ be a graph without loops. The following definitions are from Bernstein's paper. We denote by $\mathscr E_\Gamma$ the vector space over $\mathbb Q$ consisting of sets $x = (x_\alpha)$ of rational numbers $x_\alpha (\alpha \in \Gamma_0)$. We call a vector $x = (x_\alpha)$ positive (written $x > 0$) if $x \neq 0$ and $x_\alpha \geq 0$ for all $\alpha \in \Gamma_0$.

We denote by $B$ the quadratic form on the space $\mathscr E_\Gamma$ defined by the formula $B(x) = \underset{\alpha \in \Gamma_0}{\sum} x_\alpha^2 - \underset{\ell \in \Gamma_1}{\sum} x_{\gamma_1 (\ell)} x_{\gamma_2 (\ell)}$, where $x = (x_\alpha)$, and $\gamma_1 (\ell)$ and $\gamma_2 (\ell)$ are the ends of the edge $\ell$. We denote by $< , >$ the corresponding symmetric bilinear form.

For each $\beta \in \Gamma_0$ we denote by $\sigma_\beta$ the linear transformation in $\mathscr E_\Gamma$ defined by the formula $(\sigma_\beta x)_\gamma = x_\gamma$ for $\gamma \neq \beta$, $(\sigma_\beta x)_\beta = - x_\beta + \sum_{\ell \in \Gamma^\beta} x_{\gamma} (\ell)$, where $\gamma(\ell)$ is the end-point of the edge $\ell$ other than $\beta$.

We denote by $W$ the semigroup of transformations of $\mathscr E_\Gamma$ generated by the $\sigma_\beta$ ($\beta \in \Gamma_0$). $W$ is related to the Weyl group and $\sigma_\beta$ is often called the reflection.

For each $\beta \in \Gamma_0$ we denote by $\overline{\beta}$ the vector in $\mathscr E_\Gamma$ such that $(\overline{\beta})_\alpha = 0$ for $\alpha \neq \beta$ and $(\overline{\beta})_\beta = 1$.

\begin{lemma}
$1)$ If $\alpha, \beta \in \Gamma_0, \alpha \neq \beta$, then $< \overline{\alpha} , \overline{\alpha} > = 1$ and $2 < \overline{\alpha} , \overline{\beta} >$ is the negative of the number of edges joining $\alpha$ and $\beta$.
$2)$ Let $\beta \in \Gamma_0$. Then $\sigma_\beta (x) = x - 2 <\overline{\beta}, x>\overline{\beta}, \sigma_\beta^2 = 1$. In particular, $W$ is a group.
$3)$ The group $W$ preserves the integral lattice in $\mathscr E_\Gamma$ and preserves the quadratic form $B$.
$4)$ If the form $B$ is positive definite (that is, $B(x) > 0$ for $x \neq 0$), then the group $W$ is finite.
\end{lemma}

We will skip the proof and move onto more definitions.

\begin{definition}

A vector $x \in \mathscr E_\Gamma$ is called a root if for some $\beta \in \Gamma_0$, $w \in W$ we have $x = \omega \overline{\beta}$. The vectors $\overline{\beta} (\beta \in \Gamma_0)$ are called simple roots. A root $x$ is called positive if $x > 0$.

\end{definition}

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\section{Applications of Reflection Functors}

Let $(\Gamma, \Lambda)$ be a finite connected quiver. For each object $V \in \mathscr L (\Gamma, \Lambda)$ we regard the set of dimensions $\dim V_\alpha$ as a vector in $\mathscr E_\Gamma$ and denote it by $\dim V$.
We need the following unoriented graphs to state the main result of the paper, they are known as Dynkin diagrams.
\\[10pt]
\centerline{
\xymatrix{
A_n & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \dots \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet & & & (n \, \text{vertices,} \, n \geq 1)  \\
& & & & & & \bullet \[email protected]{-}[ld] \\
D_n & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \dots \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet & & & (n \, \text{vertices,} \, n \geq 4) \\
& & & & & & \bullet \[email protected]{-}[ul] \\
E_6 & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \\
& & & \bullet \[email protected]{-}[u] \\
E_7 & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \\
& & & \bullet \[email protected]{-}[u] \\
E_8 & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \\
& & & \bullet \[email protected]{-}[u] \\
}
}

\begin{theorem}
(Gabriel [2]). $1)$ If in $\mathscr L (\Gamma, \Lambda)$ there are only finitely many non-isomorphic indecomposable objects, then $\Gamma$ coincides with one of the graphs $A_n, D_n, E_6, E_7, E_8$.

$2)$ Let $\Gamma$ be a graph of one of the types $A_n, D_n, E_6, E_7, E_8$, and $\Lambda$ some orientation of it. Then in $\mathscr L (\Gamma, \Lambda)$ there are only finitely many non-isomorphic indecomposable objects. In addition, the mapping $V \mapsto \dim V$ sets up a one-to-one correspondence between classes of isomorphic indecomposable objects and positive roots in $\mathscr E_\Gamma$.
\end{theorem}

We show how reflection functors $F^+_\beta$ and $F^-_\alpha$ were used to prove part $2$ the following theorem. The following result shows that under the assumptions the quadratic form $B$ is positive definite.

\begin{proposition}

The form $B$ is positive definite for the graphs $A_n, D_n, E_6, E_7, E_8$ and only for them.

\end{proposition}

Theorem $1$ says that if $\beta$ is a sink and $V$ is an indecomposable representation of $(\Gamma, \Lambda)$, not isomorphic to $L_\beta$, then $\dim F_\beta^+ V = \sigma_\beta (\dim V)$. Part $2$ of Lemma $1$ says that $\sigma_\beta$ is an invertible

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\rfoot{}

\noindent linear transformation. Since $B$ is positive definite then $\sigma_\beta$ is an orthogonal reflection about a certain hyperplane in $\mathscr E_\Gamma$. Due to this fact, $F_\beta^+$ got its name as a reflection functor. Repeated use of Corollary $1$ implies that there is a bijection between nonisomorphic indecomposable representations of $(\Gamma, \Lambda)$ and the positive roots, given by $V \mapsto \dim V$. By part $4$ of Lemma $1$, the group $W$ is finite. Hence, the set of roots is finite and so is the set of positive roots. Therefore the set of nonisomorphic indecomposable representations is finite.

\newpage

\cfoot{}
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\lfoot{}

\begin{center}
References
\end{center}

$[1]$ I.N. Bernstein, I.M. Gel'fand, and V. A. Ponomarev, \textit{Coxeter Functors and Gabriel's} \\
\indent \indent \textit{Theorem}, Uspekhi Mat. Nauk. 28 (1973), translated in Russian Math. Surveys 28 (1973),\\
\indent \indent 17-32.

$[2]$ P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71 - 103.

$[3]$ S. Mac Lane, Homology, Springer-Verlag, 1963.

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