3 problems

Let $L$ be a finite-dimensional simple module over the relevant KLR algebra, let $d_L$ be the corresponding vector in the dual canonical basis of ${\mathbb C}[N]$, and let $b_Z$ be…

MV-basis cluster conjecture. The MV basis for $\mathbb C[N]$ contains all cluster monomials. Moreover, every element of the MV basis is $g$-pointed in each cluster.

Module-structure conjecture. For every preprojective algebra module $M$ of dimension vector $\nu$, the direct sum