7 problems

Sagan–Swanson's conjecture. The set of monomials ${\mathcal{A}}_{n,r}$ descends to a vector space basis of $SR_{n,r}$.

Finkelberg–Fourier–Littelmann conjecture. For every birational sequence $S$ and monomial ordering $>$, the monoid $\Gamma(S,>,\lambda)$ is finitely generated.

Vinberg's polyhedral-inequality conjecture. There exist a family of subsets $M_{i}\subset\{1,\dots,N\}$ and a family of elements $l_{i}\in\mathfrak{t}_{\mathbb{Z}}$ such that the s…

Vinberg's saturation conjecture. The semigroup $\Sigma$ is saturated, namely

Vinberg's finite-generation conjecture. The semigroup $\Sigma$ is finitely generated.

Let $\boldsymbol{d}:\Delta_+\rightarrow\mathbb{N}$ be a degree function for $U(\mathfrak n^-)$, and let $\mathcal{S}_{\operatorname*{gm}}$ be the set of degree functions such that,…

Monomial basis conjecture. The elements