23 problems

Flat-locus dimension conjecture. A tuple $f\in R^{\iota}_{\bf d}$ is in the flat locus of $\pi$ if and only if

Plücker-dimension conjecture. For every $M\in{\rm Rep}_{\bf d}$,

Homomorphism-dimension conjecture. Under either condition, $\dim\Gr_{\dim P}(M)=\langle\dim P,\dim I\rangle$ if and only if

Type A minimal-dimension conjecture. There exists a representation $M^2$ of dimension $\dim P$ such that $\Gr_{\dim P}(N)$, for $N\in{\rm Rep}_{\bf d}$, has dimension…

Irreducible-locus conjecture. There exists a representation $M^1$ of dimension $\dim P$ such that, for $N\in{\rm Rep}_{\bf d}$, the quiver Grassmannian $\Gr_{\dim P}(N)$ is irreduc…

Reduced scheme conjecture. For every principal quiver Grassmannian, the scheme structure defined by the quadratic relations corresponding to all paths in $Q$ is reduced.

Symplectic juggling-pattern order conjecture. The inherited combinatorial order on symplectic juggling patterns corresponds to the $G^{sp}$-orbit closure inclusion order.

Let $Q$ be a tame quiver, let ${\mathbf{r}}$ be a rank vector for $Q$, and let $\Lambda_{\mathbb C}$ be an irreducible representation of $Q$. A basis for $\Lambda_{\mathbb C}$ iden…

Let $A$ be a finite-dimensional Jacobian algebra, let $M=\operatorname{Coker}(\delta)$, and let ${{\sf V}}(M)$ denote the set of vertices of the Newton polytope of $M$. A dimension…

Let $\Gamma$ be a Dynkin quiver, let $Y$ be the associated representation variety, and let $\pi:X\to Y$ be the map whose fibres are flag versions of quiver Grassmannians. For a fie…

Let $Q$ be a quiver and let $v_1,\ldots,v_n$ be a sequence of mutations. For each vertex $i$, let $\phi_{n,i}$ be the representation on the base quiver corresponding to the last mu…

Cell-dimension torus-action conjecture. There exists a torus action on $\operatorname{Gr}_{\mathbf e}(P_{m+1}^V)$ such that the dimension of the cell labeled by $(S_H,S_V)$ is…

Torus-action conjecture. The torus action on

Cell decomposition conjecture. Every quiver Grassmannian attached to an exceptional representation of any quiver admits a cell decomposition; in particular, this should hold for qu…

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…

Let $Q$ be a Dynkin quiver, let $M$ be a representation of $Q$, and let $Gr_{\mathbf{e}}(M)$ be the associated quiver Grassmannian. A cellular decomposition means an $\alpha$-parti…

Let $M$ be a representation of a Dynkin quiver and let $Gr_{\mathbf{e}}(M)$ denote its quiver Grassmannian of subrepresentations with dimension vector $\mathbf{e}$. Property (textb…

Bistable Grassmannian conjecture. The bistable Grassmannian equals the whole Grassmannian:

Let $T$ be a tree extension of $S$, and let $M$ be a $T$-module such that $M_\alpha$ is an isomorphism for all arrows $\alpha$ in $T-S$. Let $M_S$ be the restriction of $M$ to $S$,…

Let $Q$ be a forest and let $M$ be a $Q$-module with ordered basis ${\mathcal B}$ such that, for every arrow $\alpha$ of $Q$, the matrix $M_\alpha$ is a block matrix … where $I$ is…

Let $M$ be a representation of a Dynkin quiver and let ${\bf e}$ be a dimension vector. For a generic subrepresentation type $[N]$ of $M$ in dimension ${\bf e}$, write…

Let $Q$ be an acyclic quiver, and let quiver Grassmannians be the varieties parametrizing subrepresentations of representations of $Q$. A counting polynomial for such a quiver Gras…

Let $Q$ be an affine quiver of affine type, let $lambda displaystylein mathbb P^10(Q)$, and let $E$ be a quasi-simple module in an exceptional tube. For a dimension vector…