[[Quiver representation]]
# Dimension vector of a quiver representation

Let $W : \underline{\Gamma} \to \cat{FinVect}_{\mathbb{K}}$ be a [[quiver representation]].
The **dimension vector** $\dim_{*} W \in \mathbb{N}_{0}^{\Gamma V}$ of $W$ is the function defined by #m/def/quiv
$$
\begin{align*}
(\dim_{*} V)(x) = \dim V(x)
\end{align*}
$$
where the notation is by analogy to [[graded dimension]].
We call any vector in $\mathbb{N}_{0}^{\Gamma V} \sube \mathbb{Q}^{\Gamma V}$ a dimension vector for $\Gamma$,
since one can always

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