Author: Eiko

Tags: quiver variety, quiver, dimension, euler characteristic, cartan form, euler form

Time: 2024-10-01 04:50:39 - 2024-10-01 04:52:07 (UTC)

Euler Form

Given two representations $V,W$, consider the following Euler characteristic of the derived Hom (total Hom)

$$\begin{array}{rl}⟨V,W⟩& :=\chi (R{\mathrm{Hom}}^{\bullet }(V,W))\\ & =\dim \mathrm{Hom}(V,W)-\dim {\mathrm{Ext}}^1(V,W)\\ & =\sum _i{v}_i{w}_i-\sum _{\alpha :s\alpha \to t\alpha }{v}_{s\alpha }{w}_{t\alpha }.\end{array}$$

which sometimes is called the Euler form or the Cartan form.

There is also a symmetric Euler form with round brackets

$$\begin{array}{rl}(V,W)& =⟨V,W⟩+⟨W,V⟩\\ & =2\sum _i{v}_i{w}_i-\sum _{i\leftrightarrow j}{v}_i{w}_j\\ & =\sum _i{v}_i{w}_i(2-2\sum _{i\to i}1)-\sum _{i\ne j}{v}_i{w}_j\sum _{i\leftrightarrow j}1\end{array}$$

Dimension of Representation Space

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

$$\begin{array}{rl}\dim \mathfrak{M}(Q,v,w)& =\dim \mathrm{Rep}(\bar{Q})+v\cdot w-\dim G_v-\dim (G_v/{\mathbb{C}}^{\times })\\ & =2⟨v,v⟩+v\cdot w-2v\cdot v+1\\ & =\end{array}$$