We wish to compute what \(\mathrm{Ext}^1(V, W)\) means in the case of quiver representations. First we will use the following exact sequence
\[0 \to \bigoplus_{t\leftarrow s} P_t\otimes(\leftarrow)\otimes V_s \to \bigoplus_t P_t\otimes V_t \to V \to 0\]
Apply the functor \(\mathrm{Hom}(,W)\) to get the long exact sequence
\[\begin{align*} 0 &\to \mathrm{Hom}(V, W) \to \bigoplus_t \mathrm{Hom}(P_t\otimes V_t, W) \to \bigoplus_{t\leftarrow s} \mathrm{Hom}(P_t\otimes V_s, W) \\ &\to \mathrm{Ext}^1(V, W) \to \bigoplus_t \mathrm{Ext}^1(P_t\otimes V_t, W) \end{align*}\]
Let’s compute various terms in this sequence.
First, we have
\[\begin{align*} R\mathrm{Hom}_Q(P_t\otimes_k V_t, W) &= R\mathrm{Hom}_Q(P_t\otimes_k^L V_t, W) \\ &= R\mathrm{Hom}_k(V_t, R\mathrm{Hom}_Q(P_t, W)) \\ &= R\mathrm{Hom}_k(V_t, \mathrm{Hom}_Q(P_t, W)) \\ &= R\mathrm{Hom}_k(V_t, W_t). \end{align*}\]
We have \(\mathrm{Ext}^1_Q(P_t\otimes V_t, W) = \mathrm{Ext}^1_k(V_t, W_t) = 0\) (since vector spaces only have split extensions). Thus the sequence ends with \(\mathrm{Ext}^1(V, W)\).
Next, we have
\[\begin{align*} \mathrm{Hom}(P_t\otimes_k V_t, W) &= \mathrm{Hom}_k(V_t, \mathrm{Hom}_Q(P_t, W)) \\ &= \mathrm{Hom}_k(V_t, W_t). \end{align*}\]
Finally, we have
\[\begin{align*} \mathrm{Hom}(P_t\otimes_k V_s, W) &= \mathrm{Hom}_k(V_s, \mathrm{Hom}_Q(P_t, W)) \\ &= \mathrm{Hom}_k(V_s, W_t). \end{align*}\]
This means that the sequence we get is
\[\begin{align*} 0 &\to \mathrm{Hom}(V, W) \to \bigoplus_t \mathrm{Hom}_k(V_t, W_t) \to \bigoplus_{s\rightarrow t} \mathrm{Hom}_k(V_s, W_t) \\ &\to \mathrm{Ext}^1(V, W) \to 0. \end{align*}\]
The map \(\bigoplus_t \mathrm{Hom}_k(V_t,W_t)\to \bigoplus_{s\to t} \mathrm{Hom}_k(V_s, W_t)\) is given by the action of the arrows of the quiver,
\[ (f_t)_t \mapsto (a_{s\to t}\circ f_s - f_t\circ a_{s\to t})_{s\to t}.\]
It gives us a concrete computation to the Euler characteristic
\[\dim \mathrm{Hom}(V, W) - \dim \mathrm{Ext}^1(V, W) = \sum_t \dim \mathrm{Hom}_k(V_t, W_t) - \sum_{s\to t} \dim \mathrm{Hom}_k(V_s, W_t).\]
It tells us, to form an extension of \(V\) by \(W\), we need some extension data \(\mathrm{Hom}_k(V_s, W_t)\) for each arrow \(s\to t\). Any extension is represented by a collection of such data.
Any two such data are equivalent iff they differ by the image of a fake morphism \(\bigoplus_t \mathrm{Hom}_k(V_t, W_t)\), which acts conjugately on both sides of the extension data. We call it a fake morphism because it does not necessarily commute with the arrows of the quiver.
Note that those fake morphisms that are actually morphisms, maps trivially into the extension data since they commute with the arrows of the quiver. This tells us how to compute the dimension of \(\mathrm{Ext}^1(V, W)\).
The sequence reduces to
\[\begin{align*} 0 &\to \mathrm{End}_Q(V) \to \mathrm{End}_k(V) \to \mathrm{Rep}(Q, V) \\ &\to \mathrm{Ext}^1(V, V) \to 0. \end{align*}\]
Let \([V^x]\) denotes the \(\mathrm{GL}(V)\)-orbit of certain representation \(V^x\) in the representation space \(x\in \mathrm{Rep}(Q,V)\), then
\[\begin{align*} T_x[V^x] &= T_e\mathrm{GL}(V) / T_e \mathrm{Stab}(V^x)\\ &= \mathrm{End}_k(V) / T_e \mathrm{Aut}_Q(V^x)\\ &= \mathrm{End}_k(V) / \mathrm{End}_Q(V^x). \end{align*}\]
The normal space can be computed as (using the exact sequence)
\[\begin{align*} N_x[V^x] &= T_x\mathrm{Rep}(Q,V) / T_x[V^x]\\ &= \mathrm{Rep}(Q,V)/(\mathrm{End}_k(V) / \mathrm{End}_Q(V^x))\\ &= \mathrm{Ext}^1(V^x, V^x). \end{align*}\]
Therefore a point in maximum open orbit will have trivial self extension \(\mathrm{Ext}^1(V^x, V^x) = 0\).
Let \(Q\) be a quiver, we have an Euler form defined on the \(K\)-group of the representation category, \(K(\mathrm{Rep}(Q))\)
\[\begin{align*} \langle V, W\rangle_Q &:= \chi(R\mathrm{Hom}(V,W)) \\ &= \dim \mathrm{Hom}(V, W) - \dim \mathrm{Ext}^1(V, W). \end{align*}\]
Since \(R\mathrm{Hom}\) is exact and \(\chi\) is additive on exact triangles, we have that the Euler form is bilinear on \(K(\mathrm{Rep}(Q))\), which in the quiver case is just the space of dimension vectors. In terms of dimension vectors, it can be written as
\[\langle v, w\rangle_Q = \sum v_i w_i - \sum_{s\to t} v_{s} w_{t} = v^T E w,\]
where \(E=I-A=(\delta_{ij} - a_{i\to j})\) is the Euler matrix of the quiver \(Q\).
Note that the Euler form is not symmetric in general, but we can define an associated Cartan form
\[(v,w)_Q := \langle v,w\rangle_Q + \langle w, v\rangle_Q\]
and a Cartan matrix \(C = E + E^T = 2I - A - A^T = 2I - (a_{i\leftrightarrow j})\).
Since \(E_{\overline{Q}} = I - A - A^T\), the relation between Euler form of a doubled quiver \(\overline{Q}\) and the Cartan form of the original quiver \(Q\) is
\[(,)_Q = \langle, \rangle_{\overline{Q}} + (\cdot)_Q\]
The function \(p_Q(v) = 1 - \langle v, v\rangle_Q\) is called the parameter function. It computes the number of parameters in an indecomposable representation class, when \(v\) is a positive root.
When \(v\) is not a root, we need to decompose it into sum of positive roots and compute the sum of parameters for each decomposition. The maximum of such sum is the dimension of the representation space of \(v\).
For a Nakajima quiver variety, when \(\alpha\) is a positive root, the dimension of the quiver variety is \(2p_Q(\alpha)\). The \(2\) comes from doubling the quiver.
There are three classes of roots:
\(p(\beta)=0 \Leftrightarrow \langle \beta,\beta\rangle = 1\) are called real roots.
\(p(\beta)=1 \Leftrightarrow \langle \beta,\beta\rangle = 0\) are called (isotropic) imaginary roots.
\(p(\beta)>1 \Leftrightarrow \langle \beta,\beta\rangle < 0\) are called (non-isotropic imaginary) roots.
The definition of reflection functors can be found here Gabriel-Theorem.
On dimension vectors, if \(e_i\) is loop-free vertex, \(p(e_i) = 0\) is a real root. The reflection functor acts on dimension vectors as a reflection
\[s_i : \mathbb{Z}^Q \to \mathbb{Z}^Q, \quad v \mapsto v - (v, e_i) e_i.\]
If we have a map \(f:V\to V\), we can form the dual map \(f^*: V^*\leftarrow V^*\) defined as \(f^* = (\circ f)\). When we compute the dual of a reflection map \(s_i^*\), we have
\[s_i^*(\lambda)(v) = \lambda(s_i(v)) = \lambda(v) - (v,e_i)\lambda_i.\]
We conclude that
There is also a dual reflection \(r_i:(\mathbb{Z}^Q)^*\to (\mathbb{Z}^Q)^*\) defined by
\[(r_i\lambda)(v) = \lambda(v) - (v,e_i)\lambda(e_i)\]
\[(r_i \lambda)_j = \lambda_j - (e_j, e_i)\lambda_i.\]