Irregular graphical Deligne–Simpson conjecture

Choose a marking of each conjugacy class $\breve{\mathcal{C}}_i\subset \operatorname{GL}(V_i)$ for all $i\in I$. Gluing legs to obtain a supernova graph $\widehat{\Gamma}$ with node set $\widehat{I}$ determines parameters $q,d$ and a Kac–Moody root system in $\mathbb{Z}^{\widehat{I}}$, hence the positive roots. Let $(\,\ )$ denote the associated symmetric bilinear form, and write $q^d$ and $q^{d_i}$ for the corresponding multiplicative parameters. Irregular graphical Deligne–Simpson conjecture. The graphical Deligne–Simpson problem for the graph $\Gamma$ and conjugacy classes $\breve{\mathcal{C}}$ admits a solution if and only if $d$ is a positive root, $q^d=1$, and, whenever $d=d_1+d_2+\cdots$ is a nontrivial sum of positive roots with $q^{d_1}=q^{d_2}=\cdots=1$, one has

$$\Delta(d)>\Delta(d_1)+\Delta(d_2)+\cdots,$$

where

$$\Delta(d)=2-(d,d).$$

This is the proposed irregular analogue of Crawley-Boevey's criterion for the tame Deligne–Simpson problem; the tame conjecture was announced as proved, whereas the corresponding irregular criterion is presented here as a conjecture.