6 problems

The geometric interpretation conjecture. The function $\chi_1$ is well-defined and

The allowable-path determinant conjecture. For every such Kostant picture,

The determinant conjecture. Every cluster monomial is an MV-polynomial that is naturally expressible as a determinant: if $\psi$ is a cluster monomial, then there is a Kostant pict…

Anderson–Kogan conjecture. The cluster monomials are contained in the MV basis.

Let $\mathbold{G}$ be a complex reductive group, let $\lambda$ be a dominant coweight lying in the coroot lattice, and let $W$ be the Weyl group of $\mathbold{G}$, canonically iden…

Mirkovic–Vilonen cycle and affine-paving conjecture. For every $x\in\mathscr{X}$, there exists $w\in W$ such that $w\cdot\mathscr{X}(\mathrm{Ec}(x))$ is a Mirkovic–Vilonen cycle. M…