Given two representations \(V,W\), consider the following Euler characteristic of the derived Hom (total Hom)
\[\begin{align*} \langle V, W\rangle &:= \chi(R\mathrm{Hom}^\bullet(V,W)) \\ &= \dim\mathrm{Hom}(V,W) - \dim \mathrm{Ext}^1(V,W)\\ &= \sum_i v_i w_i - \sum_{\alpha: s\alpha\to t\alpha} v_{s\alpha} w_{t\alpha}. \end{align*}\]
which sometimes is called the Euler form or the Cartan form.
There is also a symmetric Euler form with round brackets
\[\begin{align*} (V,W) &= \langle V, W\rangle + \langle W, V\rangle \\ &= 2\sum_i v_i w_i - \sum_{i\leftrightarrow j} v_i w_j\\ &= \sum_i v_i w_i \left(2 - 2\sum_{i\to i}1\right) - \sum_{i\neq j} v_i w_j \sum_{i\leftrightarrow j} 1\\ \end{align*}\]
Given any quiver \(Q\), the dimension of its representation space is given by
\[ \dim\mathrm{Rep}(Q) = \sum_{\alpha} v_{s\alpha} w_{t\alpha}.\]
Recall that the quiver variety associated to \(Q\) with dimension vector \(v\) and framing \(w\) is \(\mathfrak{M}(Q,v,w)\), which is obtained by first doubling the quiver and take the kernel of moment map, then use GIT quotient.
Therefore, its dimension is computed by
\[\begin{align*} \dim \mathfrak{M}(Q,v,w) &= \dim\mathrm{Rep}(\overline{Q}) + v\cdot w - \dim G_v - \dim (G_v/\mathbb{C}^\times) \\ &= 2\langle v, v\rangle + v\cdot w - 2v\cdot v + 1\\ &= \end{align*}\]