13 problems

Let $\mathscr{T}_{3,n}\subset \mathbb{C}[\operatorname{Gr}(3,n)]$ denote the set of invariants of all arborizable, indecomposable, non-elliptic semistandard $\operatorname{SL}_3$ w…

Web immanants and web invariants are functions associated with webs for a Grassmannian, and the basis webs are indexed by standard Young tableaux. Fraser–Lam–Le's web duality conje…

Let $r \geq 3$ and let $\mathscr{S} \subset \Pi(n,d,r)$ denote the collection of set partitions that can be made noncrossing by applying at most $r-3$ adjacent transpositions. For…

Character–web conjecture. If $\operatorname{ch}(T)$ is a cluster monomial, then

Character–web conjecture. If $\operatorname{ch}(T)$ is a cluster monomial, then

Let $\mathbb{C}[\operatorname{Gr}^{\sim}(3,n)]$ be the coordinate ring of the affine cone over the Grassmannian. A web invariant is called indecomposable, non-elliptic, or arboriza…

Forest-presentation conjecture. The web invariant $z$ is a cluster monomial if and only if $z=[D]$ for some forest diagram $D$.

Tree-presentation conjecture. The cluster variables in $R_\sigma(V)$ are distinguished from other web invariants by possessing a particular kind of alternative presentation; inform…

Three-term non-cluster conjecture. Then $z$ is not a cluster variable.

Three-term cluster conjecture. Then $z$ and $z'$ are cluster variables, and the displayed relation is an exchange relation.

Web-invariant flag conjecture. If the product of any two of them is a web invariant, then the product of all of them is a web invariant.

Cluster-monomial web-basis conjecture. All cluster monomials are web invariants.

Indecomposable-web conjecture. All cluster variables are indecomposable web invariants.