Author: Eiko

Tags: geometry, algebraic geometry, quiver variety, representation theory

Time: 2024-10-01 06:01:20 - 2025-01-09 10:33:06 (UTC)

Canonical Decomposition

Define $R_{\lambda ,\theta }^{+}=\{\alpha \in R^{+}:\lambda \cdot \alpha =\theta \cdot \alpha =0\}$, this root set is expected to be the set of indecomposable representations.

So naturally, $\mathbb{N}{R}_{\lambda }^{+}$ is the set of dimension vectors that have representations.

Consider a subset of roots ${\mathrm{\Sigma }}_{\lambda ,\theta }\subset R_{\lambda ,\theta }^{+}$ defined as

$${\mathrm{\Sigma }}_{\lambda ,\theta }=\{\alpha \in R_{\lambda ,\theta }^{+}:p(\alpha )>\sum p({\beta }^{(i)})\text{ for all }\alpha =\sum n_i{\beta }^{(i)}\in R_{\lambda ,\theta }^{+}\},$$

these are expected to be the dimension vectors that have simple / semistable representations.

Write

$$\alpha =n_1{\sigma }^{(1)}+\cdots +n_k{\sigma }^{(k)},$$

It is a canonical decomposition if

• ${\sigma }^{(i)}$s are distinct

• any other decomposition of $\alpha$ into roots in ${\mathrm{\Sigma }}_{\lambda ,\theta }$ is a refinement of this decomposition.

Properties

• The canonical decomposition gives isomorphism $${\mathfrak{M}}_{\lambda }(\alpha ,\theta )\cong \prod _i{\mathrm{Sym}}^{n_i}({\mathfrak{M}}_{\lambda }({\sigma }^{(i)},\theta ))$$

• ${\mathfrak{M}}_{\lambda }(\alpha ,\theta )$ admits projective symplectic resolution iff each ${\mathfrak{M}}_{\lambda }({\sigma }^{(i)},\theta )$ admits projective symplectic resolution.

• There exists a $\theta$-stable representation of $\mathrm{Rep}({\mathrm{\Pi }}^{\lambda },\alpha )$ iff $\alpha \in {\mathrm{\Sigma }}_{\lambda ,\theta }$.

• The irreducible components of $\mathrm{Rep}({\mathrm{\Pi }}_{\lambda },v)$ are in bijection with the set of decompositions of $v$ such that the equality $p(v)=\sum p(v^{(i)})$ holds, whose dimension will be

  $$1+2A_{Q}v\cdot v-v\cdot v.$$

Smoothness and Polystable

Polystable

• A representation is polystable iff it is a direct sum of stable representations.

• A representation $x$ is canonically $\theta$-polystable if $x=x_1\oplus \cdots \oplus x_k$ where each $x_i$ matches $\dim x_i={\beta }^{(i)}$ is $\theta$-stable and $x_i\ne x_j$ unless ${\beta }^{(i)}={\beta }^{(j)}$ is a real root, i.e. $p({\beta }^{(i)})=0$.

Smoothness

• A point $x\in {\mathfrak{M}}_{\lambda }(\alpha ,\theta )$ belongs to the smooth locus iff it is canonically $\theta$-polystable.

• The quiver variety ${\mathfrak{M}}_{\lambda }(\alpha ,\theta )$ is smooth iff

  • each ${\sigma }^{(i)}$ is minimal,

  • ${\sigma }^{(i)}$ isotropic $\Rightarrow n_i=1$.

References

Gwyn Bellamy and Travis Schedler, Symplectic Resolutions of Quiver Varieties.

Victor Ginzburg, Lectures on Nakajima’s quiver varieties.