7 problems

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…

Vertex-preservation conjecture. The map $\phi_\lambda$ induces a bijection

Standard string-polytope conjecture. The polytope $Q_{\underline{w_0}^\mathrm{std}}(\lambda)$ is a lattice polytope if and only if at least one of the following holds:…

Alexeev–Brion's integrality conjecture. The string polytope $\Delta_{\bf i}(\lambda)$ is integral for every reduced word $\bf i$ of the longest element in the Weyl group and every…

Integrality conjecture for standard string polytopes. The string polytope $\mathcal{Q}_{\underline{w_0}^{\mathrm{std}}}(\lambda)$ is a lattice polytope if and only if at least one…

Alexeev–Brion conjecture. For every reduced decomposition $\underline{w_0}$ and every $\lambda\in\Lambda^+$, the string polytope $\mathcal{Q}_{\underline{w_0}}(\lambda)$ is a latti…

Fix a Borel subgroup $B\subset G$ and a sequence $I=(i_1,\ldots,i_\ell)$ of simple roots defining the Bott–Samelson variety $R_I$. Let $P_v(\mathcal L)$ be the Okounkov body of a l…