9 problems

Let $A$ be the set of planar bipartite graphs embedded in a disk satisfying the following conditions: there are $n$ boundary vertices, all in the same part of a bipartition; every…

Let $\mathcal{T}(n,n)$ be the set of standard Young tableaux of shape $(n,n)$, let $\{v_T:T\in\mathcal{T}(n,n)\}$ and $\{w_S:S\in\mathcal{T}(n,n)\}$ be the tableau and web bases of…

Russell–Tymoczko positivity conjecture. Every transition-matrix entry $a_{TM}$ is nonnegative.

Transition-coefficient positivity conjecture. For every tableau $T'$ with corresponding web $w'$, one has

$\mathfrak{sl}_3$-web conjecture. Similar results to those established for the webs considered in the paper should hold for webs for $\mathfrak{sl}_3$.

Positivity and support conjecture. The entries $c_w^{w'}$ of the transition matrix are nonnegative, and an entry is nonzero precisely when the corresponding directed path exists in…

Consider the Kazhdan–Lusztig basis and the basis of reduced webs, and let the change-of-basis matrix express one basis in terms of the other. Unitriangularity conjecture. The chang…

In the $\mathfrak{sl}_3$ case, consider the dual canonical basis and the reduced web basis, with the former restricted to the relevant web space. Kuperberg's conjecture. The restri…

Defining-relations conjecture. The relations given for the web basis of $\mathrm{SL}(4)$ are defining relations; that is, they generate all relations among the corresponding web di…