Originally posted here 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!

Originally posted here 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!

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!

Originally posted here 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!