6 problems

Positivity conjecture. For every $b\in\mathbf{C}^\imath$, all coefficients $f_c(v)$, $g_{b',c}(v)$ and $h_{c,c'}(v)$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$. In particular…

Let $T \in \operatorname{SSYT}(k,[n])$ be a tableau with no frozen factors. Write $T=S^{\cup r}$ with $r$ maximal, where $S=\underline{T}$ has no further common frozen factor. A po…

Leclerc's conjecture. The product has the form

The dual canonical basis conjecture. For every $T\in {\rm SSYT}(k-1,[k])$,

Let $w$ be a Weyl group element, let ${\mathsf{A_q}}[N_{-}(w)]$ be the corresponding quantum unipotent subgroup, and let $\mathsf{B}^{{\mathsf{up}}}(w)$ be its dual canonical basis…

Let ${\mathfrak{g}}$ be a Kac--Moody algebra with symmetrizable Cartan datum, let $\mathsf{U_q}^{-}$ be the negative part of its quantized enveloping algebra, and let…