Author: Eiko

Tags: stratification, quiver variety, geometry

Time: 2024-11-21 16:06:28 - 2024-11-26 21:17:18 (UTC)

Decomposition

• Canonical decomposition gives a factor into products of (symmetric powers of) quiver varieties.

  $$\mathcal{M}(\alpha )\cong \prod _i{\mathrm{Sym}}^{n_i}\mathcal{M}({\alpha }^{(i)})$$

• When we have any decomposition (not necessarily canonical) we have a map by taking direct sum of the decomposed representations

  $$\prod _i\mathcal{M}\left({\alpha }^{(i)}\right)\stackrel{\oplus }{\to }\mathcal{M}\left(\sum _i{\alpha }^{(i)}\right)$$

  which is not necessarily an embedding.

  whose image is a stratum we want to understand. Actually it is the normalisation of the image (closure of stratum).

Questions To Look At

• Things are already known about the quiver side. What does the stratums look like and described in terms of hilbert schemes?