13 problems

Tropical tensor-product conjecture. The tropicalization of $F$ is an isomorphism of torsion-free Kashiwara crystals

Schubert-cell crystal conjecture. For every $w\in W$,

Saturated-valuation conjecture. Under these assumptions, such a strongly positive unipotent bicrystal exists and the valuation-induced map identifies the normal crystal of…

Perfect-basis realization conjecture. Such a based module and linear map exist.

Injectivity conjecture. For every connected component of $\mathcal{B}_{f,\Theta}$, the restriction of $\tilde {\bf f}_w$ is injective.

Weyl-invariant generation conjecture. The subfield of $\mathbb{C}(z'_{j,i})$ fixed by this birational action is generated by the reduced $Q$-invariants and shape invariants.

Q-invariant and barred P-invariant generation conjecture. The field ${\rm Inv}_{R\overline{R}}$ is generated by the $Q$-invariants, barred $P$-invariants, and shape invariants. Eve…

Q-invariant and shape-invariant generation conjecture. The field ${\rm Inv}_{R\overline{e}}$ is generated by the $Q$-invariants and shape invariants. Moreover, every element of…

Demazure-crystal expansion conjecture. There exists a reduced expression of the longest Weyl-group element $\mathbf i=(i_1,\ldots,i_N)$ and a sign $p=(p_{i,j})_{i\ne j}$ such that,…

Let $I$ be the index set of the root system, let $\mathbf i=(i_1,\ldots,i_N)$ be a reduced expression of the longest Weyl-group element, and let $p=(p_{i,j})_{i\ne j}$ be the assoc…

KNO conjecture. For any $i\in I\setminus\{0\}$, there exists a unique variety $X$ endowed with a positive $\mathfrak g$-geometric crystal structure and a rational map…

Coherent-family limit conjecture. The crystal ${\mathcal UD}(\chi,T,\theta)$ is the limit of a coherent family of perfect crystals of type $\TY(G,1,2)$ in the cited work.

Let $G$ and $\mathfrak g=\langle \mathfrak t,e_i,f_i\rangle_{i\in I}$ be the affine Kac–Moody group and algebra associated with a generalized Cartan matrix, and let $B^-_w$ be the…