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
\[\prod_{i}\mathcal{M}\left(\alpha^{(i)}\right) \xrightarrow{\oplus} \mathcal{M}\left(\sum_{i}\alpha^{(i)}\right)\]
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).