23 problems

Let $\mathfrak{g}$ be an affine Lie algebra with Dynkin index set $I$, and let $i\in I\setminus\{0\}$. A positive geometric crystal for $\mathfrak{g}$ is a quadruple … where $X$ is…

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 $\mathfrak g$ be the affine Kac–Moody group and algebra associated with a generalized Cartan matrix, let $I$ be its index set, and let $W(\varpi_i)$ be the fundamental…

Berenstein–Kazhdan's conjecture. For every algebraic $\ell$-dimensional representation $\rho$ of $G^\vee$, there exists a $W$-equivariant algebro-geometric distribution…

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

Geometric tensor-product conjecture. The inverse of $F$ is an isomorphism of positive decorated geometric 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.

Let $\mathfrak g$ be an affine Kac–Moody algebra with index set $I$, distinguished node $0$, Weyl group $W$, extended Weyl group $\widetilde W$, and Langlands dual $\mathfrak g^L$.…

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I$, let $k\in I\setminus\{0\}$ be a Dynkin index, and let $\mathfrak{g}^L$ be the Langlands dual of $\mathfrak{g}$. A po…

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…

Positive geometric crystal existence conjecture. For each affine Lie algebra $\mathfrak{g}$ and each Dynkin index $k\in I\setminus\{0\}$, there exists a positive geometric crystal

Positive geometric crystal conjecture. For each affine Lie algebra $\mathfrak{g}$ and each Dynkin node $i\in I\setminus\{0\}$, there exists a positive geometric crystal whose ultra…

Uniqueness conjecture. The point $y=\overline\sigma(x)$ is the unique solution 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…

KNO conjecture. The affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect…

Let $G$ be a semisimple algebraic group, let $U^-$ be the unipotent radical of an opposite Borel subgroup, and let $X_w=\overline X_w$ be the Schubert variety associated with a Wey…