10 problems

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

The coefficient conjecture. For $k<d$, the coefficient of $t^k$ in $\mathcal P_{\mathbf p}(\mathbf n)$ is $0$; for $k>d$, the coefficient depends on the choice of $\mathbf n$.

The cluster-monomial coefficient conjecture. The coefficient of $t^d$ in $\mathcal P_{\mathbf p}(\mathbf n)$ is independent of the choice of $\mathbf n$ representing $Y$, and defin…

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.

Anderson's fusion conjecture. This fusion product should match the multiplication in the coordinate ring $\mathbb{C}[N]$ of $N$.

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…

Let $G$ be the reductive group and $T$ its maximal torus, let $M\in\Lambda(d)$ be a module over the preprojective algebra, and let $X(M)$ be the associated cycle. The quiver Grassm…

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…