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Author: Danika Van Niel
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36\title{Reflection Functors in the Representation Theory of Quivers}
37\author{Danika Van Niel}
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49\begin{document}
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61$\textbf{Abstract:}$
62The 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.
63\newpage
64$\textbf{Executive Summary:}$
65
66\cfoot{}
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69
70The 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.
71
72Consider 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.
73
74We showed that $\mathscr L$$(\Gamma,\Lambda) satisfies the following conditions and therefore is a category: 75\begin{enumerate} 76\item The composition of morphisms is a morphism and the composition is associative 77\item For all morphisms \phi: (U,f) \to (V,g), \, 1_{(V,g)}\phi = \phi 1_{(U,g)} = \phi 78\end{enumerate} 79 80Reflection 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. 81 82We show that F_\beta^+: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\beta\Lambda) satisfies the following conditions and therefore is a functor: 83 84\begin{enumerate} 85\item F_\beta^+ (1_{(U,f)}) = 1_{(X,r)} 86 87\item F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi)) 88 89\end{enumerate} 90 91Similarly we can show that F_\alpha^- is a functor. 92 93After 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. 94 95We 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. 96 97Now 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. 98 99 100 101\newpage 102 103\cfoot{} 104\lfoot{\thepage} 105\rfoot{} 106 107\renewcommand{\headrulewidth}{0pt} 108 109\tableofcontents 110 111\newpage 112 113\cfoot{} 114\rfoot{\thepage} 115\lfoot{} 116 117\section{Introduction} 118This project is about representations of quivers which is an area of mathematics that uses methods of linear algebra, combinatorics and category theory. \\ 119Recall some necessary definitions from linear algebra. \\ 120\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. 121 122\centerline{ 123\xymatrix{ 124Ker\,\psi \ar[r]^\kappa & V \ar[r]^\psi & W \\ 125U \ar[u]^\xi \ar[ru]_\phi 126} 127} 128 129\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. 130 131\centerline{ 132\xymatrix{ 133V \ar[r]^{\psi} & W \ar[r]^{\pi} \ar[dr]_{\phi} & Coker\,\psi \ar[d]^{\xi} \\ 134&& U 135} 136} 137 138\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. 139\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. 140\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. 141 142\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. 143 144Define \Gamma as a finite connected graph with the set of vertices \Gamma_0 and the set of edges \Gamma_1. Fix an 145 146\newpage 147 148\cfoot{} 149\lfoot{\thepage} 150\rfoot{} 151 152\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). 153 154With 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$$(\Gamma,\Lambda) is any collection (V,f) of finite dimensional vector spaces V_\alpha \, (\alpha \in \Gamma_0) and linear mappings f_\ell (\ell \in \Gamma_1). 155There is a particular representation where all the vector spaces are zero and all the maps are the zero maps, called 0. A \textbf{morphism} \phi: (V,f) \to (W,g) is a collection of linear mappings \phi_\alpha: V_\alpha \to W_\alpha (\alpha \in \Gamma_0) such that for each edge \ell \in \Gamma_1 the following diagram 156 157\centerline{ 158\xymatrix{ 159V_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]^{\phi_{\alpha(\ell)}} & V_{\beta(\ell)} \ar[d]^{\phi_{\beta(\ell)}} \\ 160W_{\alpha(\ell)} \ar[r]_{g_\ell} & W_{\beta(\ell)} 161} 162} 163 164\noindent is commutative, that is, \phi_{\beta(\ell)} f_\ell = g_\ell \phi_{\alpha(\ell)}. The objects of \mathscr L$$(\Gamma,\Lambda)$ are called representations of the quiver $(\Gamma,\Lambda)$ and the category $\mathscr L$$(\Gamma,\Lambda) is called the category of representations of (\Gamma,\Lambda). 165 166\par We define the law of composition for morphisms as follows. Let \phi: (U,f) \to (V,g) and \psi: (V,g) \to (W,h) be morphisms where \phi = (\phi_\alpha)_{\alpha \in \Gamma_0} and \psi = (\psi_\alpha)_{\alpha \in \Gamma_0}. Then \psi \circ \phi: (U,f) \to (W,h) is given by (\psi \circ \phi)_\alpha = \psi_\alpha \circ \phi_\alpha. 167\\[11pt] 168\noindent Define the \textbf{identity morphism} 1_{(V,f)} for an object (V,f) by 1_{(V,f)} = (1_{V_\alpha})_{\alpha \in \Gamma_0}. 169We prove that \mathscr L$$(\Gamma,\Lambda)$ is a category in the next section.
170
171%---------------------------------------------
172\newpage
173
174\cfoot{}
175\rfoot{\thepage}
176\lfoot{}
177
178\section{The Category of Representations}
179
180\noindent We show that $\mathscr L$$(\Gamma,\Lambda) satisfies the following conditions and therefore is a category: 181\begin{enumerate} 182\item The composition of morphisms is a morphism and the composition is associative 183\item For all morphisms \phi: (U,f) \to (V,g), \, 1_{(V,g)}\phi = \phi 1_{(U,g)} = \phi 184\end{enumerate} 185 186For any objects (U,f), (V,g), and (W,h) in \mathscr L$$(\Gamma,\Lambda)$, let $\phi: (U,f) \to (V,g)$ and \\ $\psi: (V,g) \to (W,h)$ be morphisms. Then we have a commutative diagram,
187
188\centerline{
189\xymatrix{
190U_{\alpha(\ell)} \ar[d]_{\phi_{\alpha(\ell)}} \ar[r]^{f_\ell} & U_{\beta(\ell)} \ar[d]^{\phi_{\beta(\ell)}} \\
191V_{\alpha(\ell)} \ar[d]_{\psi_{\alpha(\ell)}} \ar[r]^{g_\ell} & V_{\beta(\ell)} \ar[d]^{\psi_{\beta(\ell)}} \\
192W_{\alpha(\ell)} \ar[r]^{h_\ell} & W_{\beta(\ell)}
193}
194}
195
196\noindent that is
197\begin{equation} \label{1}
198\phi_{\beta(\ell)}f_\ell = g_\ell\phi_{\alpha(\ell)}
199\end{equation}
200and $\psi_{\beta(\ell)} g_\ell = h_\ell \psi_{\alpha(\ell)}$. Then
201
202\begin{center}
203$\psi_{\beta(\ell)} \phi_{\beta(\ell)} f_\ell = \psi_{\beta(\ell)} (\phi_{\beta(\ell)} f_\ell) = \psi_{\beta(\ell)} (g_\ell \phi_{\alpha(\ell)}) =$ \\
204$(\psi_{\beta(\ell)} g_\ell) \phi_{\alpha(\ell)} = (h_\ell \psi_{\alpha(\ell)}) \phi_{\alpha(\ell)} = h_\ell \psi_{\alpha(\ell)} \phi_{\alpha(\ell)}$
205\end{center}
206\noindent which shows that $\psi \circ \phi : (U,f) \to (W,h)$ is a morphism, that is the diagram
207
208\centerline{
209\xymatrix{
210U_{\alpha(\ell)} \ar[d]_{[\psi \circ \phi]_{\alpha(\ell)}} \ar[r]^{f_\ell} & U_{\beta(\ell)} \ar[d]^{[\psi \circ \phi]_{\beta(\ell)}} \\
211W_{\alpha(\ell)} \ar[r]^{h_\ell} & W_{\beta(\ell)}
212}
213}
214
215\noindent commutes.
216\\[12pt]
217    We have shown that the composition of morphisms is well-defined.
218\\[12pt]
219Suppose that $\phi$ and $\psi$ are as above, and $\xi: (W,h) \to (Y,j)$ is a morphism in $\mathscr L$$(\Gamma,\Lambda) where \xi = (\xi_\alpha), \, \alpha \in \Gamma_0 220Then, using the associativity of composition of linear mappings we get 221\begin{center} 222[(\xi \circ \psi) \circ \phi]_\alpha = (\xi \circ \psi)_\alpha \circ \phi_\alpha = (\xi_\alpha \circ \psi_\alpha) \circ \phi_\alpha = \xi_\alpha \circ (\psi_\alpha \circ \phi_\alpha) = \xi_\alpha \circ (\psi \circ \phi)_\alpha = [\xi \circ (\psi \circ \phi)]_\alpha. 223\end{center} 224Therefore, (\xi \circ \psi) \circ \phi = \xi \circ (\psi \circ \phi). We have shown the composition of morphisms is associative. Thus \mathscr L$$(\Gamma,\Lambda)$ satisfies the first property.
225\\[12pt]
226For a morphism $\phi : (U,f) \to (V,g)$ as above, we have
227
228\begin{center}
229$[1_{(V,g)} \circ \phi]_\alpha = (1_{(V,g)})_\alpha \circ \phi_\alpha = \phi_\alpha$ \, and \,
230$[\phi \circ 1_{(U,f)}]_\alpha = \phi_\alpha \circ (1_{(U,f)})_\alpha = \phi_\alpha$.
231\end{center}
232
233\newpage
234
235\cfoot{}
236\lfoot{\thepage}
237\rfoot{}
238
239\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.
240
241\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 $K$. \\
242\centerline{
243$\phi = (\phi_\alpha)_{\alpha \in \Gamma_0} \, , \, \psi = (\psi_\alpha)_{\alpha \in \Gamma_0}$}
244We define $\phi + \psi$ by
245
246\centerline{
247$(\phi + \psi)_\alpha = \phi_\alpha + \psi_\alpha$}
248
249\noindent and, for $c \in K$ we define $c\phi$ by
250
251\centerline{
252$(c\phi)_\alpha = c\phi_\alpha$.
253}
254
255Referencing 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.
256
257The verification that $c\phi$ is a morphism is similar.
258
259In 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.
260
261A 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.
262
263\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.
264
265%-----------------------------------------
266\newpage
267
268\cfoot{}
269\rfoot{\thepage}
270\lfoot{}
271
272\section{Reflection Functors}
273
274We present the following facts from the "Coxeter Functors and Gabriel's Theorem" paper written by Bernstein, Gel'fand, and Ponomarev. \\
275
276For 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 $\ell \in \Gamma^\alpha$.
277
278\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 $\ell \in \Gamma_1$.
279
280\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$$(\Gamma,\sigma_\alpha \Lambda). These functors send an indecomposible representation to either an indecomposible representation or to zero. We construct such a functor for each vertex \alpha at which all the edges have the same direction. 281 282We will prove that F_\beta^+ is a functor in section 3.1, and that F_\alpha^- is a functor in section 3.2. 283 284%------------------------------------------------------------------------------------------------------- 285 286\subsection{A Positive Reflection Functor} 287 288Suppose that the vertex \beta of the graph \Gamma is a sink with respect to the orientation \Lambda. From an object (U,f) in \mathscr L$$(\Gamma,\Lambda)$ we construct a new object $F_\beta^+(U,f) = (X,r)$ in $\mathscr L$$(\Gamma,\sigma_\beta\Lambda). 289\par Namely, we put X_\gamma = U_\gamma for \gamma \neq \beta. To construct X_\beta we consider all the edges \ell_1, \ell_2, \ldots , \ell_k that end at \beta (that is, all edges of \Gamma^\beta). We denote by X_\beta the subspace in the direct sum \underset{i = 1}{\overset{k}{\oplus}} U_{\alpha(\ell_i)} consisting of the vectors u = (u_1, \ldots, u_k) (here u_i \in U_{\alpha(\ell_i)}) for which f_{\ell_i}(u_1) + \ldots + f_{\ell_k}(u_k) = 0. In other words, if we denote by h_U the mapping h_U: \underset{i = 1}{\overset{k}{\oplus}} U_{\alpha(\ell_i)} \to U_\beta defined by the formula h_U (u_1, u_2, \ldots, u_k) = f_{\ell_1}(u_1) + \ldots + f_{\ell_k}(u_k), then X_\beta = Ker\,h_U. 290\par We now define the mappings r_{\ell_j}. For \ell_j \notin \Gamma^\beta we put r_{\ell_j} = f_{\ell_j}. If \ell = \ell_j \in \Gamma^\beta, then r_{\ell_j} is defined as the composition of the natural embedding \kappa_U: X_\beta \to \oplus U_{\alpha(\ell_i)} of X_\beta in \oplus U_{\alpha(\ell_i)} and the projection \pi_{U,{\alpha(\ell_j)}}: \oplus U_{\alpha(\ell_i)} \to U_{\alpha(\ell_j)} of the sum \oplus U_{\alpha(\ell_i)} onto the term U_{\alpha(\ell_j)} = X_{\alpha(\ell_j)}. In other words, r_{\ell_j} = \pi_{U,{\alpha(\ell_j)}} \kappa_U . We note that on all edges \ell_j \in \Gamma^\beta the orientation has been changed, that is, the resulting object (X,r) belongs to \mathscr L$$(\Gamma,\sigma_\beta\Lambda)$.
291Let $\phi = (\phi_\alpha): (U,f) \to (V,g)$ be a morphism in $\mathscr L$$(\Gamma,\Lambda), let (X,r) = F^+_\beta(U,f) and (Y,s) = F^+_\beta (V,g). We construct F^+_\beta(\phi) = \xi = (\xi_\alpha)_{\alpha \in \Gamma_0}: (X,r) \to (Y,s). 292If \alpha \neq \beta, then X_\alpha = U_\alpha, Y_\alpha = V_\alpha, and we set \xi_\alpha = \phi_\alpha: U_\alpha \to V_\alpha. To construct \xi_\beta: X_\beta \to Y_\beta, we consider the following diagram of vector spaces and linear maps 293\begin{equation} 294\xymatrix{ 295X_\beta \ar[r]^{\kappa_U} \ar[d]_{\xi_\beta} & \oplus^k_{i = 1}U_{\alpha(\ell_i)} \ar[r]^{h_U} \ar[d]^{\oplus\phi_{\alpha(\ell_i)}} & U_\beta \ar[d]^{\phi_\beta} \\ 296Y_\beta \ar[r]^{\kappa_V} & \oplus^k_{i = 1}V_{\alpha(\ell_i)} \ar[r]^{h_V} & V_\beta 297} 298\end{equation} 299\noindent where X_\beta = Ker\,h_U, Y_\beta = Ker\,h_V, and \kappa_U and \kappa_V are the inclusion maps. It is easy to verify that the right square of the diagram commutes. 300 301\begin{center} 302\phi_\beta h_U = h_V(\oplus^k_{i=1}\phi_{\alpha(\ell_i)}) 303\end{center} 304 305\newpage 306 307\cfoot{} 308\lfoot{\thepage} 309\rfoot{} 310 311\noindent Since h_V( \underset{i = 1}{\overset{k}{\oplus}} \phi_{\alpha(\ell_i)})\kappa_U = \phi_\beta h_U\kappa_U = \phi_\beta0 = 0, the universal property of the kernel (see Introduction) says that there exists a unique k-linear map \xi_\beta: X_\beta \to Y_\beta satisfying \kappa_V\xi_\beta = (\underset{i = 1}{\overset{k}{\oplus}} \phi_{\alpha(\ell_i)})\kappa_U. 312This finishes the construction of \xi = F^+_\beta(\phi). We now verify that it is a morphism in \mathscr L$$(\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
313
314\begin{center}
315$\xi_{\alpha(\ell_j)} \pi_{U,{\alpha(\ell_j)}}(u_1, \dots, u_k) = \xi_{\alpha(\ell_j)}(u_j) = \phi_{\alpha(\ell_j)}(u_j)$ and \\
316$\pi_{V_{\alpha(\ell_j)}}[\oplus \phi_{\alpha(\ell_i)}](u_1, \dots, u_k) = \pi_{V_{\alpha(\ell_j)}}(\phi_{\alpha(\ell_1)}(u_1), \dots, \phi_{\alpha(\ell_k)}(u_k)) = \phi_{\alpha(\ell_j)}(u_j)$. Hence \\
317$\xi_{\alpha(\ell_j)} \pi_{U,\alpha(\ell_j)} = \pi_{V,\alpha(\ell_j)} [\oplus \phi_{\alpha(\ell_i)}]$ and we have \\
318$\xi_{\alpha(\ell_j)} r_{\ell_j} =\xi_{\alpha(\ell_j)} \pi_{U,{\alpha(\ell_j)}}\kappa_U = \pi_{V,\alpha(\ell_j)}[\oplus \phi_{\alpha(\ell_i)}]\kappa_U = \pi_{V,\alpha(\ell_j)} \kappa_V \xi_\beta = s_{\ell_j} \xi_\beta$.
319\end{center}
320
321For each edge $\ell \in \Gamma_1$ not incident to $\beta$, we have $\alpha(\ell) \neq \beta$, $\beta(\ell) \neq \beta$, so
322
323\centerline{
324\xymatrix{
325U_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]_{\phi_{\alpha(\ell)}} & U_{\beta(\ell)} \ar[d]^{\phi_{\alpha(\ell)}} \\
326V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)}
327}
328}
329
330\noindent is a commutative diagram because $\phi: (U,f) \to (V,g)$ is a morphism. Hence the above construction yields the commutative diagram
331
332\centerline{
333\xymatrix{
334X_{\alpha(\ell)} \ar[r]^{r_\ell} \ar[d]_{\xi_{\alpha(\ell)}} & X_{\beta(\ell)} \ar[d]^{\xi_{\beta(\ell)}} \\
335Y_{\beta(\ell)} \ar[r]^{s_\ell} & Y_{\beta(\ell)}
336}
337}
338
339\noindent as required.
340
341We show that $F_\beta^+: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\beta\Lambda)$ satisfies the following conditions and therefore is a functor:
342
343\begin{enumerate}
344\item $F_\beta^+ (1_{(U,f)}) = 1_{(X,r)}$
345
346\item $F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi))$
347
348\end{enumerate}
349
350As 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: $\xi_\alpha = 1_{X_\alpha}$, $\alpha \in \Gamma_0$.
351
352If $\alpha \neq \beta$, then $\xi_\alpha = \phi_\alpha$, but $\phi_\alpha = 1_{U_\alpha} = 1_{X_\alpha}$ since $\alpha \neq \beta$.
353
354To 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
355
356\centerline{
357\xymatrix{
358X_\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}} \\
359X_\beta \ar[r]^{\kappa_U} & \oplus U_{\alpha(\ell_i)} \ar[r]^{h_U} & U_\beta
360}
361}
362
363\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 $\xi_\beta = 1_{X_\beta}$.
364
365\noindent Hence, $F^+_\beta(1_{(U,f)}) = 1_{(X,r)}$. \\[12pt]
366
367\newpage
368
369\cfoot{}
370\rfoot{\thepage}
371\lfoot{}
372
373\noindent Now we check if $F_\beta^+ (\psi\phi) = (F_\beta^+ (\psi))(F_\beta^+(\phi))$. \\
374\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. 375 376\noindent Set 377 378\begin{center} 379F_\beta^+(\phi) = \xi = (\xi_\alpha)_{\alpha \in \Gamma_0} \\ 380F_\beta^+(\psi) = \zeta = (\zeta_\alpha)_{\alpha \in \Gamma_0} \\ 381F_\beta^+(\psi \phi) = \theta = (\theta_\alpha)_{\alpha \in \Gamma_0} 382\end{center} 383 384\noindent We want to show that \theta_\alpha = \zeta_\alpha \xi_\alpha, \alpha \in \Gamma_0. 385 386\noindent a) For \alpha \neq \beta 387 388\begin{center} 389\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 390\end{center} 391 392\noindent b) For \alpha = \beta we set X_\beta = Ker\,h_U, Y_\beta = Ker\,h_V, and Z_\beta = Ker\,h_W 393 394\centerline{ 395\xymatrix{ 396X_\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} \\ 397Y_\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} \\ 398Z_\beta \ar[r]^{\kappa_W} & \oplus^k_{i=1} W_{\alpha(\ell_i)} \ar[r]^{h_W} & W_\beta 399} 400} 401 402\noindent By (2) the above diagram commutes so 403 404\begin{center} 405[\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 406\end{center} 407 408By (2), the diagram below commutes. \\ 409 410\centerline{ 411\xymatrix{ 412X_\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} \\ 413Z_\beta \ar[r]^{\kappa_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\alpha(\ell_i)} \ar[r]^{h_W} & W_\beta 414} 415} 416 417\noindent We have 418 419\begin{center} 420 421[\oplus (\psi \phi)_{\alpha(\ell_i)}] \kappa_U = \kappa_W \theta_\beta 422 423\end{center} 424 425So 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. 426 427It 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. 428 429%% ---------------------------------------------------------------------------------------------- 430 431\newpage 432 433\cfoot{} 434\lfoot{\thepage} 435\rfoot{} 436 437\subsection{A Negative Reflection Functor} 438 439Suppose 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$$(\Gamma,\sigma_\alpha\Lambda). 440\par Namely, we put X_\gamma = U_\gamma for \gamma \neq \alpha. 441\par Next we consider all the edges \ell_1, \ell_2, \ldots , \ell_k that start at \alpha (that is, all edges of \Gamma^\alpha). We denote by \widetilde{h}_U the mapping \widetilde{h}_U : U_\alpha \to \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} defined by the formula \widetilde{h}_U(u) = (f_{\ell_1}(u), \ldots, f_{\ell_k}(u)), and set X_\alpha = Coker\, \widetilde{h}_U = \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)}/Im\,\widetilde{h}_U. Denote by \pi_U : \oplus U_{\beta(\ell_i)} \to X_\alpha the canonical map. 442\par We now define the mappings r_\ell. For \ell \notin \Gamma^\alpha we put r_\ell = f_\ell. If \ell = \ell_j \in \Gamma^\alpha, then r_{\ell_j} is defined as the composition of the natural embedding \kappa_{U, \ell_j} : U_{\beta(\ell_j)} \to \underset{i = 1}{\overset{k}{\oplus}} U_{\beta(\ell_i)} and the canonical map \pi_{U}: \oplus U_{\beta(\ell_i)} \to X_\alpha. In other words, r_{\ell_j} = \pi_U \kappa_{U,\beta(\ell_j)}. We note that on all edges \ell \in \Gamma^\alpha the orientation has been changed, that is, the resulting object (X,r) belongs to \mathscr L$$(\Gamma,\sigma_\alpha\Lambda)$. Let $\phi = (\phi_\beta): (U,f) \to (V,g)$ be a morphism in $\mathscr L$$(\Gamma,\Lambda), let (X,r) = F^-_\alpha(U,f) and (Y,s) = F^-_\alpha (V,g). We construct F^-_\alpha(\phi) = \xi = (\xi_\beta)_{\beta \in \Gamma_0}: (X,r) \to (Y,s). 443If \beta \neq \alpha, then X_\beta = U_\beta, Y_\beta = V_\beta, and we set \xi_\beta = \phi_\beta: U_\beta \to V_\beta. To construct \xi_\alpha: X_\alpha \to Y_\alpha, we consider the following diagram of vector spaces and linear maps 444\begin{equation} 445\xymatrix{ 446U_\alpha \ar[r]^{\widetilde{h_U}} \ar[d]_{\phi_\alpha} & \oplus^k_{i = 1}U_{\beta(\ell_i)} \ar[r]^{\pi_{U}} \ar[d]^{\oplus\phi_{\beta(\ell_i)}} & X_\alpha \ar[d]^{\xi_\alpha} \\ 447V_\alpha \ar[r]^{\widetilde{h_V}} & \oplus^k_{i = 1}V_{\beta(\ell_i)} \ar[r]^{\pi_{V}} & Y_\alpha 448} 449\end{equation} 450\noindent where X_\alpha = Coker\,\widetilde{h_U}, Y_\alpha = Coker\,\widetilde{h_V}, and \pi_{U} and \pi_{U} are the canonical maps. It is easy to verify that the left square of the diagram commutes. 451 452\begin{center} 453\widetilde{h_V} \phi_\alpha = (\oplus^k_{i=1}\phi_{\beta(\ell_i)}) \widetilde{h_U} 454\end{center} 455 456\noindent Since \pi_V (\oplus \phi_{\beta(\ell_i)}) \widetilde{h}_U = \pi_V \widetilde{h}_V \phi_\alpha = 0, the universal property of the cokernel (see Introduction) says that there exists a unique k-linear map \xi_\alpha: X_\alpha \to Y_\alpha satisfying \pi_V(\oplus \phi_{\beta(\ell_i)}) = \xi_\alpha \pi_U. 457This finishes the construction of \xi = F^-_\alpha(\phi). We now verify that it is a morphism in \mathscr L$$(\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
458
459$$[\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_j)} = \kappa_{V, \beta(\ell_j)} \xi_{\beta(\ell_j)}.$$
460
461\noindent Indeed
462
463$$[\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_i)}(u_j) = [\oplus \phi_{\beta(\ell_i)}](0, \ldots, u_j, \ldots, 0) = (0, \ldots, \phi_{\beta(\ell_j)}(u_j), \ldots, 0)$$
464
465\noindent and
466
467
468$$\kappa_{V, \beta(\ell_j)} \xi_{\beta(\ell_i)}(u_j) = \kappa_{V, \beta(\ell_j)} (\phi_{\beta(\ell_i)} (u_j)) = (0, \ldots , \phi_{\beta(\ell_j)}(u_j), \ldots, 0).$$
469
470\noindent Therefore
471
472\begin{center}
473$\xi_\alpha r_{\ell_j} = \pi_V [\oplus \phi_{\beta(\ell_i)}] \kappa_{U, \beta(\ell_j)} = \pi_V \kappa_{V,\beta(\ell_j)} \xi_{\beta(\ell_j)} = s_{\ell_j} \xi_{\beta(\ell_j)}$.
474
475\end{center}
476
477\newpage
478
479\cfoot{}
480\rfoot{\thepage}
481\lfoot{}
482
483For each edge $\ell \in \Gamma_1$ not incident to $\alpha$, we have $\beta(\ell) \neq \alpha$, $\alpha(\ell) \neq \alpha$, so
484
485\centerline{
486\xymatrix{
487U_{\alpha(\ell)} \ar[r]^{f_\ell} \ar[d]_{\phi_{\alpha(\ell)}} & U_{\beta(\ell)} \ar[d]^{\phi_{\alpha(\ell)}} \\
488V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)}
489}
490}
491
492\noindent is a commutative diagram because $\phi: (U,f) \to (V,g)$ is a morphism. Hence the above construction yield the commutative diagram
493
494\centerline{
495\xymatrix{
496X_{\alpha(\ell)} \ar[r]^{r_\ell} \ar[d]_{\xi_{\alpha(\ell)}} & X_{\beta(\ell)} \ar[d]^{\xi_{\beta(\ell)}} \\
497Y_{\beta(\ell)} \ar[r]^{s_\ell} & Y_{\beta(\ell)}
498}
499}
500
501\noindent as required.
502
503We show that $F_\alpha^-: \mathscr L (\Gamma,\Lambda) \to \mathscr L (\Gamma,\sigma_\alpha\Lambda)$ satisfies the following conditions and therefore is a functor:
504
505\begin{enumerate}
506\item $F_\alpha^- (1_{(U,f)}) = 1_{(X,r)}$
507
508\item $F_\alpha^- (\psi\phi) = (F_\alpha^- (\psi))(F_\alpha^-(\phi))$
509
510\end{enumerate}
511
512As 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: $\xi_\beta = 1_{X_\beta}$, $\beta \in \Gamma_0$.
513
514If $\beta \neq \alpha$, then $\xi_\beta = \phi_\beta$, but $\phi_\beta = 1_{U_\beta} = 1_{X_\beta}$ since $\beta \neq \alpha$.
515
516To 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
517
518\centerline{
519\xymatrix{
520U_\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} \\
521U_\alpha \ar[r]^{\widetilde{h}_U} & \oplus U_{\beta(\ell_i)} \ar[r]^{\pi_U} & X_\alpha
522}
523}
524
525\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 $\xi_\alpha = 1_{X_\alpha}$.
526
527\noindent Hence, $F^-_\alpha(1_{(U,f)}) = 1_{(X,r)}$. \\[12pt]
528
529\noindent Now we check if $F_\alpha^- (\psi\phi) = (F_\alpha^- (\psi))(F_\alpha^-(\phi))$. \\
530\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. 531 532\noindent Set 533 534\begin{center} 535F_\alpha^-(\phi) = \xi = (\xi_\beta)_{\beta \in \Gamma_0} \\ 536F_\alpha^-(\psi) = \zeta = (\zeta_\beta)_{\beta \in \Gamma_0} \\ 537F_\alpha^-(\psi \phi) = \theta = (\theta_\beta)_{\beta \in \Gamma_0} 538\end{center} 539 540\noindent We want to show that \theta_\beta = \zeta_\beta \xi_\beta, \beta \in \Gamma_0. 541 542\noindent a) For \beta \neq \alpha 543 544\newpage 545 546\cfoot{} 547\lfoot{\thepage} 548\rfoot{} 549 550\begin{center} 551\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 552\end{center} 553 554\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 555 556\centerline{ 557\xymatrix{ 558U_\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} \\ 559V_\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} \\ 560W_\alpha \ar[r]^{\widetilde{h}_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\beta(\ell_i)} \ar[r]^{\pi_W} & Z_\alpha 561} 562} 563 564\noindent By (3) the above diagram commutes so 565 566\begin{center} 567 \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 568\end{center} 569 570By (3), the diagram below commutes. \\ 571 572\centerline{ 573\xymatrix{ 574U_\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} \\ 575W_\alpha \ar[r]^{\widetilde{h}_W} & \underset{i = 1}{\overset{k}{\oplus}} W_{\beta(\ell_i)} \ar[r]^{\pi_W} & Z_\alpha 576} 577} 578 579\noindent We have 580 581\begin{center} 582 583\pi_W [\oplus (\psi \phi)_{\beta(\ell_i)}] = \theta_\alpha \pi_U 584 585\end{center} 586 587So 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. 588 589It 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. 590 591%----------------------------------------------------------------------------------------------- 592 593\subsection{Properties of Reflection Functors} 594 595Let (\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. 596 597\begin{theorem} 598 5991) 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
600\noindent $a) \, V \approx L_\beta$ and $F^+_\beta V = 0$. \vskip.05in
601\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
602
603\newpage
604\cfoot{}
605\rfoot{\thepage}
606\lfoot{}
607
608\begin{center}
609
610$\dim F^+_\beta(V)_\gamma = \dim V_\gamma \, \text{ for} \, \gamma \neq \beta$, \\
611$\dim F^+_\beta(V)_\beta = -\dim V_\beta + \underset{\ell \in \Gamma^\beta}{\Sigma} \dim V_{\alpha(\ell)}$.
612
613\end{center}
614
615$2)$ 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
616\noindent $a) \, V \approx L_\alpha$ and $F^-_\alpha (V) = 0$. \vskip.05in
617\noindent $b) \, F^-_\alpha (V)$ is an indecomposable representation, $F^+_\alpha F^-_\alpha (V) = V$, and
618\begin{center}
619
620$\dim F^-_\alpha (V)_\gamma = \dim V_\gamma$ for $\gamma \neq \alpha$, \\
621$\dim F^-_\alpha (V)_\alpha = -\dim V_\alpha + \underset{\ell \in \Gamma^\alpha}{\Sigma}$ $\dim V_{\beta(\ell)}$.
622
623\end{center}
624
625\end{theorem}
626
627
628Proof. 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. 629 630\begin{equation} \label{4} 631\xymatrix{ 632Y_\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} \\ 633& & & & V_\beta 634} 635\end{equation} 636 637Here 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 638 639\centerline{ 640\xymatrix{ 641Z_{\alpha(\ell)} \ar[d]_{(i_V^\beta)_{\alpha(\ell)}} \ar[r]^{t_\ell} & Z_{\beta(\ell)} \ar[d]^{(i_V^\beta)_{\beta(\ell)}} \\ 642V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_{\beta(\ell)} 643} 644} 645 646\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 647 648\begin{center} 649 650(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)} 651 652\end{center} 653 654\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. 655 656\centerline{ 657\xymatrix{ 658V_{\alpha(\ell)} \ar[r]^{\pi_Y \kappa_{V, \alpha(\ell)}} \ar[d]_{Id} & Z_\beta \ar[d]^{(i_V^\beta)_\beta} \\ 659V_{\alpha(\ell)} \ar[r]^{g_\ell} & V_\beta 660} 661} 662 663\newpage 664 665\cfoot{} 666\lfoot{\thepage} 667\rfoot{} 668 669Similarly, 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. 670 671\begin{lemma} 6721) F^\pm_\alpha (V_1 \oplus V_2) = F^\pm_\alpha(V_1) \oplus F^\pm_\alpha(V_2). \\ 6732) p_V^\alpha is an epimorphism and i^\beta_V is a monomorphism. \\ 6743) 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). \\ 6754) 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. \\ 6765) If the representation V has the form F_\beta^- W (F^+_\alpha W respectively), then i^\beta_V (p^\alpha_V) is an isomorphism. \\ 6776) 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). 678\end{lemma} 679 680Say how define direct sum in category, then use fact about categories then use that they are additive functors to prove 1. 681 682For 2 we have that for it to be an whatever all of it's parts also have to be an whatever. 683 684Proof. 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$$(\Gamma, \Lambda)$, and that $\pi_j \iota_j = 1_{(V_j, g_j)}, j = 1,2,$ as well as $\iota_1 \pi_1 + \iota_2 \pi_2 = 1_{(V_1 \oplus V_2, h)}$.
685
686Since we know that $\mathscr L$$(\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.
687
688\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$.
689\end{proposition}
690Proof: 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
691
692\begin{center}
693$F\pi_j Fi_j = F(\pi_ji_j) = F(1_{A_j}) = 1_{F A_j}$ \\
694$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}$
695\end{center}
696\vskip.1in
697
698$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.
699\par Similarly it is easy to verify that $p_V^\alpha$ is an epimorphism.
700\vskip.1in
701
702\newpage
703
704\cfoot{}
705\rfoot{\thepage}
706\lfoot{}
707
708$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.
709
710\centerline{
711\xymatrix{
7120 \ar[r] & F_\beta^+(V)_\beta \ar[r] & \oplus_{i = 1}^{k} V_{\alpha(\ell_i)} \ar[r] & V_\beta \ar[r] & 0
713}
714}
715
716\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.
717
718\par Likewise, if $p_V^\alpha$ is an isomorphism then the equations from Theorem $1$ part $2b)$ holds.
719
720\vskip.1in
721
722$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$.
723
724\vskip.1in
725
726$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.
727
728\par Similarly, the statement regarding $p_V^\alpha$ holds.
729
730\vskip.1in
731
732$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$.
733We 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. \\ 734Case I). V = V/Im\, i_V^\beta. Then V_\gamma = 0 for \gamma \neq \beta and, because V is indecomposable, V \approx L_\beta. \\ 735Case 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. 736So we have shown that the representation F^+ _\beta (V) is indecomposable. We can similarly prove (2) of Theorem 1. 737 738\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. 739 740\begin{corollary} 741 742Let (\Gamma, \Lambda) be an oriented graph and \alpha_1, \alpha_2, \ldots, \alpha_k a sink sequence. 7431) 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 $L_{\alpha_i} \in \mathscr L (\Gamma, \sigma_{\alpha_{i - 1}} \ldots \sigma_{\alpha_{1}} \Lambda$)). \\
744$2)$ Let $V \in \mathscr L (\Gamma, \Lambda)$ be an indecomposable representation, and \\
745\centerline{
746$F^+_{\alpha_k} \cdot \ldots \cdot F^+_{\alpha_1} (V) = 0$
747}
748Then for some $i$ \\
749\centerline{
750$V \approx F^-_{\alpha_1} \cdot \ldots \cdot F^-_{\alpha_{i - 1}} (L_{\alpha_i})$.
751}
752
753\end{corollary}
754
755
756%----------------------------------------------------------------------------------------------
757
758\newpage
759
760\cfoot{}
761\lfoot{\thepage}
762\rfoot{}
763
765
766Let $\Gamma$ 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 $\alpha \in \Gamma_0$.
767
768We 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 $\gamma_1 (\ell) 769$ and $\gamma_2 (\ell)$ are the ends of the edge $\ell$. We denote by $< , >$ the corresponding symmetric bilinear form.
770
771For 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 $\beta$.
772
773We 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.
774
775For 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 $(\overline{\beta})_\beta = 1$.
776
777\begin{lemma}
778$1)$ 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 $\beta$.
779$2)$ 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.
780$3)$ The group $W$ preserves the integral lattice in $\mathscr E_\Gamma$ and preserves the quadratic form $B$.
781$4)$ If the form $B$ is positive definite (that is, $B(x) > 0$ for $x \neq 0$), then the group $W$ is finite.
782\end{lemma}
783
784We will skip the proof and move onto more definitions.
785
786\begin{definition}
787
788A 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 $x > 0$.
789
790\end{definition}
791
792
793
794%----------------------------------------------------------------------------------------------
795\newpage
796
797\cfoot{}
798\rfoot{\thepage}
799\lfoot{}
800
801\section{Applications of Reflection Functors}
802
803Let $(\Gamma, \Lambda)$ 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 $\dim V$.
804We need the following unoriented graphs to state the main result of the paper, they are known as Dynkin diagrams.
805\\[10pt]
806\centerline{
807\xymatrix{
808A_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)  \\
809& & & & & & \bullet \[email protected]{-}[ld] \\
810D_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) \\
811& & & & & & \bullet \[email protected]{-}[ul] \\
812E_6 & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \\
813& & & \bullet \[email protected]{-}[u] \\
814E_7 & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \[email protected]{-}[r] & \bullet \\
815& & & \bullet \[email protected]{-}[u] \\
816E_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 \\
817& & & \bullet \[email protected]{-}[u] \\
818}
819}
820
821\begin{theorem}
822(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 $A_n, D_n, E_6, E_7, E_8$.
823
824
825
826
827$2)$ 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 $\mathscr E_\Gamma$.
828\end{theorem}
829
830We 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.
831
832\begin{proposition}
833
834The form $B$ is positive definite for the graphs $A_n, D_n, E_6, E_7, E_8$ and only for them.
835
836\end{proposition}
837
838Theorem $1$ 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
839
840\newpage
841
842\cfoot{}
843\lfoot{\thepage}
844\rfoot{}
845
846\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.
847
848
849
850
851\newpage
852
853\cfoot{}
854\rfoot{\thepage}
855\lfoot{}
856
857\begin{center}
858References
859\end{center}
860
861$[1]$ I.N. Bernstein, I.M. Gel'fand, and V. A. Ponomarev, \textit{Coxeter Functors and Gabriel's} \\
862\indent \indent \textit{Theorem}, Uspekhi Mat. Nauk. 28 (1973), translated in Russian Math. Surveys 28 (1973),\\
863\indent \indent 17-32.
864
865$[2]$ P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71 - 103.
866
867$[3]$ S. Mac Lane, Homology, Springer-Verlag, 1963.
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873\end{document}
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