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

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\begin{center}
A Capstone Project Submitted in Partial Fulfillment of the\\
Requirements of the Ren\'ee Crown University Honors Program at \\
Syracuse University
\end{center}

\begin{center}
Candidate for Bachelor of Science in Mathematics from the College of Arts and Sciences \\
and Ren\'ee Crown University Honors \\
May 20
\end{center}

\indent \indent \indent \indent \indent \indent \indent \indent Honors Capstone Project in Mathematics

\indent \indent \indent \indent \indent \indent \indent \indent Capstone Project Advisor: \underline{Professor Mark Kleiner}

\indent \indent \indent \indent \indent \indent \indent \indent Capstone Project Reader: \underline{Professor Steven P. Diaz}

\indent \indent \indent \indent \indent \indent Honors Director: \underline{Stephen Kuusisto, Director}

Date:

\newpage

\maketitle

\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 a vector space $X$ is isomorphic to $V \oplus W$ if and only if there exist four linear maps $V \rightleftharpoons_{\pi_V}^{\iota_V} X \leftrightharpoons_{\pi_W}^{\iota_W} W$ satisfying $\pi_V\iota_V = 1_V$, $\pi_W\iota_W = 1_W$, and $\iota_V\pi_V + \iota_W\pi_W = 1_X$.

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

Define $\Gamma$ as a finite connected graph with the set of vertices $\Gamma_0$ and the set of edges $\Gamma_1$. Fix an 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]^{\varphi_{\alpha(\ell)}} & V_{\beta(\ell)} \ar[d]^{\varphi_{\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 $(\Gamma,\Lambda)$. \cite{1}

\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$$(\Gamma,\Lambda) is a category in the next section. %--------------------------------------------- \section{The Category of Representations} \noindent We show that \mathscr L$$(\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$$(\Gamma,\Lambda) satisfies the first property. \\[12pt] For a morphism \phi : (U,f) \to (V,g) as above, we have \begin{center} [1_{(V,g)} \circ \phi]_\alpha = (1_{(V,g)})_\alpha \circ \phi_\alpha = \phi_\alpha \, and \, [\phi \circ 1_{(U,f)}]_\alpha = \phi_\alpha \circ (1_{(U,f)})_\alpha = \phi_\alpha. \end{center} Therefore 1_{(V,g)} \circ \phi = \phi \circ 1_{(U,f)} = \phi. We have shown that \mathscr L$$(\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 \oplus_{\alpha \in \Gamma_0} [keep trying to figure out how to do underset?] Hom_K(U_\alpha, V_\alpha) is a subspace. Therefore since we know that \oplus_{\alpha \in \Gamma_0} 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 [to show the direct sum exists must show that there are the 4 morphisms from section 4 chapter 1 and that they satisfy the 5 conditions(will need commutative diagrams to prove they are morphisms, show \iota_1 and \pi_1 are morphisms then their conterparts are similar)] \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. %----------------------------------------- \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 $\oplus^k_{i=1}U_{\alpha(\ell_i)}$ 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 $h$ the mapping $h: \oplus^k_{i=1}U_{\alpha(\ell_i)} \to U_\beta$ defined by the formula $h(u_1, u_2, \ldots, u_k) = f_{\ell_1}(u_1) + \ldots + f_{\ell_k}(u_k)$, then $X_\beta =$ Ker\,$h$.
\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}

\noindent Since $h_V(\oplus^k_{i=1}\phi_{\alpha(\ell_i)})\kappa_U = \phi_\beta h_U\kappa_U = \phi_\beta0 = 0$, 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 $\kappa_V\xi_\beta = (\oplus^k_{i=1}\phi_{\alpha(\ell_i)})\kappa_U$.
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] \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}
$[\oplus(\psi\phi)_{\alpha(\ell_i)}] \kappa_U = (\oplus_{i = 1}^k \psi_{\alpha(\ell_i)}\phi_{\alpha(\ell_i)}) \kappa_U = (\oplus_{i = 1}^k \psi_{\alpha(\ell_i)}) (\oplus_{i = 1}^k \phi_{\alpha(\ell_i)}) \kappa_U = (\oplus_{i = 1}^k \psi_{\alpha(\ell_i)}) \kappa_V \xi_\beta = \kappa_W \zeta_\beta \xi_\beta$
\end{center}

By $(2)$, the diagram below commutes. \\

\centerline{
\xymatrix{
X_\beta \ar[d]_{\theta_\beta} \ar[r]^{\kappa_U} & \oplus^k_{i=1} U_{\alpha(\ell_i)}  \ar[d]_{\oplus^k_{i=1} (\psi\phi)_{\alpha(\ell_i)}} \ar[r]^{h_U} & U_\beta \ar[d]^{(\psi\phi)_\beta} \\
Z_\beta \ar[r]^{\kappa_W} & \oplus^k_{i=1} 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.

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

\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 $\widetilde{h}_U : U_\alpha \to \oplus^k_{i=1}U_{\beta(\ell_i)}$ defined by the formula $\widetilde{h}_U(u) = (f_{\ell_1}(u), \ldots, f_{\ell_k}(u))$, and set $X_\alpha =$Coker$\, \widetilde{h}_U = \oplus^k_{i = 1} U_{\beta(\ell_i)}/$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 $\kappa_{U, \ell_j} : U_{\beta(\ell_j)} \to \oplus_{i = 1}^k U_{\beta(\ell_i)}$ of $U_{\beta(\ell_j)}$ into the direct sum $\oplus U_{\beta(\ell_i)}$ 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} 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 $\beta \neq \alpha$

\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} & \oplus^k_{i=1} U_{\beta(\ell_i)}  \ar[d]_{\oplus^k_{i=1} \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} & \oplus^k_{i=1} V_{\beta(\ell_i)}  \ar[d]_{\oplus^k_{i=1} \psi_{\beta(\ell_i)}} \ar[r]^{\pi_V} & Y_\alpha \ar[d]^{\zeta_\alpha} \\
W_\alpha \ar[r]^{\widetilde{h}_W} & \oplus^k_{i=1} W_{\beta(\ell_i)} \ar[r]^{\pi_W} & Z_\alpha
}
}

\noindent By $(3)$ the above diagram commutes so

\begin{center}
$\pi_W [\oplus(\psi\phi)_{\beta(\ell_i)}] = \pi_W (\oplus_{i = 1}^k \psi_{\beta(\ell_i)}\phi_{\beta(\ell_i)}) = \pi_W (\oplus_{i = 1}^k \psi_{\beta(\ell_i)}) (\oplus_{i = 1}^k \phi_{\beta(\ell_i)}) = \zeta_\alpha \pi_V (\oplus_{i = 1}^k \phi_{\beta(\ell_i)}) = \zeta_\alpha \xi_\alpha \pi_U$
\end{center}

By $(3)$, the diagram below commutes. \\

\centerline{
\xymatrix{
U_\alpha \ar[d]_{(\psi \phi)_\alpha} \ar[r]^{\widetilde{h}_U} & \oplus^k_{i=1} U_{\beta(\ell_i)}  \ar[d]_{\oplus^k_{i=1} (\psi\phi)_{\beta(\ell_i)}} \ar[r]^{\pi_U} & X_\alpha \ar[d]^{\theta_\alpha} \\
W_\alpha \ar[r]^{\widetilde{h}_W} & \oplus^k_{i=1} 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$$(\Gamma,\Lambda) be an indecomposable representation. Then two cases are possible: \vskip.05in \noindent a) \, V \approx L_\beta and F^+_\beta V = 0. \vskip.05in \noindent b) \, F^+_\beta (V) is an indecomposable representation, F^-_\beta F^+_\beta (V) = V, and the dimensions of the spaces F^+_\beta (V)_\gamma can be calculated by the formula \begin{center} \dim F^+_\beta(V)_\gamma = \dim V_\gamma \, \text{ for} \, \gamma \neq \beta, \\ \dim F^+_\beta(V)_\beta = -\dim V_\beta + \underset{\ell \in \Gamma^\beta}{\Sigma} \dim V_{\alpha(\ell)}. \end{center} 2) If the vertex \alpha is a source, and if V \in$$\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$$(\Gamma,\Lambda) is defined. For each representation V \in \mathscr L$$(\Gamma,\Lambda)$ we construct a morphism $i^\beta_V$: $F_\beta^- F_\beta^+ (V) \to V$ in the following way.
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 $(i^\beta_V)_\beta$ we note that in the sequence of mappings $F_\beta^+ (V)_\beta =$Ker$\, h \to^{\widetilde{h}} \oplus_{\ell \in \Gamma^\beta} V_{\alpha(\ell)} \to^h V_\beta$ we have Ker$\, h =$ Im$\, \widetilde{h}$ by definition of $\beta$ being a sink; we take for $(i^\beta_V)_\beta$ the natural mapping using the first isomorhphism theorem\\
\centerline{
\xymatrix{
\oplus_{\ell \in \Gamma^\beta} (V)_{\alpha(\ell)} \ar[d] \ar[rr]^f && V_\beta \\
X/\text{Ker}\, f \ar[urr]_{\widetilde{f}}
}
}

\centerline{
$F^-_\beta F^+_\beta (V)_\beta = \oplus_{\ell \in \Gamma^\beta} V_{\alpha(\ell)}/\text{Im} \widetilde{h} = \oplus_{\ell \in \Gamma^\beta} V_{\alpha(\ell)} / \text{Ker} h \to V_\beta$
}

To verify that $i_V^\beta$ is morphism for all $\ell \in \Gamma^\beta$, $\alpha(\ell) \to^\ell \beta$ we must check if the following diagram commutes: [find a formula for $g_\ell$]

\centerline{
\xymatrix{
F^-_\beta F^+_\beta (V)_{\alpha(\ell)} \ar[r]^{g_\ell} \ar[d]^{Id} & F^-_\beta F^+_\beta (V)_\beta \ar[d]^{(i^\beta_V)_\beta} \\
V_{\alpha(\ell)} \ar[r]_{f_\ell} & V_\beta
}
}

\noindent Since $(i_V^\beta)_\beta f_\ell F^-_\beta F^+_\beta (V)_\alpha = (i_V^\beta)_\beta F^-_\beta F^+_\beta (V)_\beta = V_\beta$ by computations above, and $g_\ell$ Id $F^-_\beta F^+_\beta (V)_\alpha = g_\ell V_\alpha = V_\beta$ by definitions and computations above. Then clearly, $g_\ell$ Id $= (i^\beta_V)_\beta f_\ell$ is true.

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 bject Ker$\, p_V^\alpha$ is concentrated at $\alpha$ (that is, (Ker$\, p_V^\alpha)_\gamma = 0$ for $\gamma \neq \alpha$). The representation $V/$Im$\, p_V^\alpha$ is concentrated at $\beta$. \\
$5)$ If the representation $V$ has the form $F_\alpha^+ W$ ($F^-_\beta W$ respectively), then $p^\alpha_V$ ($i^\beta_V$) 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, $V \approx F^+_\alpha F^-_\alpha (V) \oplus$Ker$\, p_V^\alpha$).
\end{lemma}

Proof. $1), 2), 3), 4)$ and $5)$ can be verified immediately. Let us prove $6)$.
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$$(\Gamma, \Lambda), and \beta a sink vertex with respect to \Lambda. Since V \approx F^-_\beta F^+_\beta (V) \oplus V/Im\, i_V^\beta and V is indecomposable, V coincides with one of the terms. \\ Case I). V = V/Im\, i_V^\beta. Then V_\gamma = 0 for \gamma \neq \beta and, because V is indecomposable, V \approx L_\beta. \\ Case II). V = F^-_\beta F^+_\beta (V), that is, i_V^\beta is an isomorphism. Then (1.1.1) is satisfied by Lemma 1. We show that the representation W = F_\beta^+ (V) is indecomposable. For suppose that W = W_1 \oplus W_2. Then V = F^-_\beta (W_1) \oplus F^+_\beta (W_2) and so one of the terms (for example, F^-_\beta (W_2)) is 0. By 5) of Lemma 1 the morphism p_V^\beta : W \to F^+_\beta F^-_\beta (W) is an isomorphism, but p_V^\beta (W_2) \subset F^+_\beta F^-_\beta (W_2) = 0, that is, W_2 = 0. So we have shown that the representation F^+ _\beta (V) is indecomposable. We can similarly prove 2) of Theorem 1. We say that a sequence of vertices \alpha_1, \ldots, \alpha_k is a sink with respect to \Lambda if \alpha_1 is a sink with respect to \Lambda, \alpha_2 is a sink with respect to \sigma_{\alpha_1} \Lambda, \alpha_3 is a sink with respect to \sigma_{\alpha_2} \sigma_{\alpha_1} \Lambda, and so on. We define a source sequence similarly. \begin{corollary} Let (\Gamma, \Lambda) be an oriented graph and \alpha_1, \alpha_2, \ldots, \alpha_k a sink sequence. 1) For any i (1 \leq i \leq k), F^-_{\alpha_1} \cdot \ldots \cdot F^-_{\alpha_{i - 1}} (L_{\alpha_i}) is either 0 or an indecomposable representation in \mathscr L$$(\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}

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

\section{Applications of Reflection Functors}

We illustrate the application of the functors $F^+_\beta$ and $F^-_\alpha$ by the following theorem.

\printbibliography

\end{document}