13 problems

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

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.

Admissible monomial basis conjecture. For $1\leq r\leq p-1$, the set of admissible monomials of the form defined in the source, where $r_L=r$, $r_0=1$, and $s=1$, is a basis of…

Monomial basis conjecture. In the general case $p<p'$, the monomials satisfying the quadratic exclusion condition

For each $n$, let ${\mathcal B}_n$ be the recursively constructed set of monomials in ${\rm DQ}_n$, using the rules … when $i+j<n-1$, together with the diagonal rule … for…

Let $S$ be a birational sequence and let $>$ be a monomial ordering. For each dominant weight $\lambda$, let $\Gamma(S,>,\lambda)$ be the monoid of essential monomials, and define…

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

Monomial-basis conjecture. The displayed spanning set is a basis of $L(k\Lambda_0)$ for all $k\geq 1$ and $\ell\geq 1$. This extends the cases already known for type $A_1$ at all p…

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

Let $\widetilde{\mathbf C}$ be the quotient of $\mathbb{R}[\mathbf{x}]$ by the subspace of hit-polynomials for the operators $\widetilde D_k$, and let…