Modules of generalized Kronecker quivers

Let k be an algebraically closed field, and let (0) denote the 0-dimensional vector space over k. Recall that n-Kronecker quiver K(n) is just the quiver with two vertices 1, 2, and n arrows \gamma_1, \cdots, \gamma_n: 1\rightarrow 2. We use rep_k(K(n)) to denote the finite-dimensional representations of K(n) If we use mod \mathcal{K}_n to denote the finite-dimensional modules of \mathcal{K}_n, where \mathcal{K}_n=kK(n), then there is an equivalence between categories rep_k(K(n)) and mod \mathcal{K}_n. We will frequently identify these two categories. Let (T(n),\Omega) be the universal covering of K(n), that is, T(n) is an n-regular tree with a bipartite orientation \Omega. In general, we can take k-modules of the quiver T(n) as graded Kronecker modules. We use mod(T(n),\Omega) to denote the finite-dimensional graded Kronecker modules (or simply modules). In the end, we introduce two reflection functors \sigma,\sigma^-: mod(T(n),\Omega)\rightarrow mod(T(n),\sigma\Omega) concerning all sink points or all source points of T(n)_0, respectively, where \sigma\Omega is the orientation under action of \sigma. If \sigma^t(M)\neq (0) for any t\in\mathbb{Z}, we call that M is \textit{regular}, where M\in mod(T(n),\Omega) is an indecomposable module. Category mod(T(n),\Omega) admits AR-sequences and its regular components are type of \mathbb{Z}A_{\infty}. Let M\in mod(T(n),\Omega) be an indecomposable module. We call M a sink (source, or flow) module if there exists a path of maximal length in M connecting two sink points (or source points, or one sink and one source points). Hence we can define center C(M), radius r(M) of M. Moreover, a module Y\in mod(T(n),\Omega) is said to be a sink (flow or source) module if Y=\oplus_{i\in I}Y^i, and all Y^i have the same center and are indecomposable sink (flow or source) modules. Let \mathcal{D} be a regular component of mod(T(n),\Omega), and let \sigma \mathcal{D} be the corresponding component under the action of reflection functor \sigma. Let (0)\rightarrow X\rightarrow Y\rightarrow Z\rightarrow (0) be an AR-sequence in \mathcal{D}, and let b(Z) be the number of flow modules in the \sigma-orbit of Z, then \begin{enumerate} \item Y is a sink (source or flow) module if and only if \sigma Z is a sink (source or flow) module. Moreover, their radii and centers satisfy r(Y)=r(\sigma Z)+1 and C(Y)=C(\sigma Z). \item \mid \mathcal{W}(\mathcal{D}) -\mathcal{W}(\sigma \mathcal{D})\mid \leq 1 , \mathcal{W}(\mathcal{D})\geq \frac{b(\mathcal{D})+1}{2}, where \mathcal{W}(\mathcal{D}) is the width of \mathcal{D} and b(\mathcal{D})=b(Z). \end{enumerate} We define full subcategories \Inj_{n,\Omega}, \Sur_{n,\Omega} and \CR_{n,\Omega} on mod(T(n),\Omega). There exists a push-down functor \pi_\lambda: mod(T(n),\Omega)\rightarrow mod \mathcal{K}_n, and it is exact. Let M\in mod(T(n),\Omega) be an indecomposable module. Let N=\pi_{\lambda}(M), and let N be not isomorphic to a simple module. We say that M is of type A_{1,2} if there exists a path in M consisting of all the direct summands M(\gamma^j_i),M(\gamma^{j}_{i'}) for some two indexes i,i', where N(\gamma_x)=\oplus_{\pi(\gamma^j_x)=\gamma_x} M(\gamma^j_x), x\in\{i,i'\}. J. Carlson, E. Friedlander and J. Pevtsova once introduced the subclass of modules of constant rank in 2008. We say that M is of constant rank, provided \pi_\lambda(M) is of constant rank. We show that M\in\Inj_{n,\Omega}\cup\Sur_{n,\Omega} when M is of type A_{1,2} and is of constant rank. If there exists \mathcal{W}(\sigma \mathcal{D})=1 for a regular component \mathcal{D} of mod(T(n),\Omega). Then we have \begin{center} \CR_{n,\Omega}\cap \mathcal{D}= (\Inj_{n, \Omega}\cup\Sur_{n, \Omega})\cap \mathcal{D}. \end{center} Let M be a finite-dimensional module of \mathcal{K}_n, we consider its dimension vector \underline{dim} M=(\text{dim}_k M_1, \text{dim}_kM_2). Let \mathbf{F}=\{(x,y)\mid \frac{2}{n}x\leq y\leq x\}. Suppose that (x,y)\in\mathbf{F}. We construct a module X(x,y) of \mathcal{K}_n, and we prove that the module M is elementary. Suppose that \underline{dim}M=(x,y). We show that: \begin{enumerate} \item The module M is an elementary module, then x<2n. \item When x+y=n+1, the module M is elementary if and only if M is of the form X(x,y). \end{enumerate}



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