Originally posted [here](https://math.stackexchange.com/questions/1931924/construction-of-irreps-of-path-algebra-of-cyclic-quiver-classification-of-all-f) on Mathematics Stack Exchange.


Let $Q$ be a quiver with vertex set $\{1, 2, \ldots, n\}$ such that $Q$ has a single edge $i \to i + 1$, for every $i = 1, 2, \ldots, n - 1$, one edge $n \to 1$, and no other edges. In other words, $Q$ is a cyclic quiver with $n > 1$ vertices. I am looking for a proof or a precise reference for the following.


1. The construction of an irrep of $\mathbb{C}Q$ of (total) dimension $> 1$.
2. The classification of all finite dimensional irreps of $\mathbb{C}Q$ up to isomorphism.


Thanks in advance!



Construction of irreps of path algebra of cyclic quiver, classification of all finite-dimensional irreps
Every irreducible finite-dimensional representation is either $1$-dimensional (and there are exactly $n$ of them, corresponding to the vertices) or $n$-dimensional and they can be indexed by non-zero, complex numbers.


Proof: Let $V$ be an irreducible representation, $V=\bigoplus\_{i\in\mathbb{Z}/n} V\_i$ its standard decomposition given by the vertices and $f\_i : V\_i\to V\_{i+1}$ the linear maps given by the edges. Let $L:=f\_{n-1}\ldots f\_1 f\_0$ be the endomorphism of $V\_0$ given by the closed loop in the quiver.
If $V$ is finite-dimensional, then there is a eigenvector $v\_0$ of $L$ for the eigenvalue $\lambda$ (we can assume that there are eigenvectors at all by assuming $V\_0\neq 0$ wlog). Define $v\_1:=f\_0(v\_0), v\_2:=f\_1(v\_1), \ldots, v\_{n-1}:=f\_{n-2}(v\_{n-2})$.


If $\lambda=0$, then $f\_i(v\_i)=0$ for some $i$ and $\mathbb{C} v\_i$ is a one-dimensional submodule. By simplicity it is the whole of $V$ and we're in the first case.


If $\lambda\neq 0$, then all $v\_i$ are nonzero and $\sum\_i \mathbb{C}v\_i$ is a submodule. By simplicity it is all of $V$, $\{v\_i\}$ is a basis and one can easily write down all matrices of the edge elements w.r.t. this basis. Therefore this is a complete set of pairwise non-isomorphic irreducibles.




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In fact it should be possible to describe all finite-dimensional modules in terms of a Jordan normal form of $L$. More abstractly there should be something close to a Morita equivalence between the quiver algebra $\mathbb{C}Q$ and $e\_0 \mathbb{C}Q e\_0$ which is just the polynomial ring $\mathbb{C}[L]$. Note that the irreducible finite-dimensional representations of $\mathbb{C}[L]$ are parametrized by complex numbers. There seems to be some sort of geometry here lurking in the background where $Irr(\mathbb{C}[L])$ is just the affine line and $Irr(\mathbb{C}Q)$ is the affine line with $n$-fold origin. Maybe some algebraic geometer can shed some light on this observation.


