151 problems

Let $U_{q}^{7}(\mathfrak{g})$ be an affine quantum group without the degree operator, and let $W^{r,s}$ be a Kirillov–Reshetikhin module, where $r$ is a node of the Dynkin diagram…

Rigged configuration virtualization conjecture. For all $\mathfrak{g}$ of symmetrizable type, there exists a simply-laced type $\widehat{\mathfrak{g}}$ and diagram folding $\phi$ w…

Crystal structure conjecture. The localization $\widetilde{\mathscr C}_w$ possesses a crystal $\mathbb B(\widetilde{\mathscr C}_w)$.

Let $L{A}$ be the subring of $\mathbb{C}(v)$ consisting of rational functions without a pole at $v=0$. Let $L{L}$ be the $L{A}$-submodule of $U^+_v(Lg{g})$ generated by the prescri…

Let $B^{r,s}$ be a Kirillov–Reshetikhin crystal associated with a Kirillov–Reshetikhin module $W^{r,s}$, where $r$ is a node of the affine Dynkin diagram other than $0$ and…

Let $B^{r,s}$ be a Kirillov–Reshetikhin crystal, with $r\in I_0$ and $s\in\mathbb{Z}_{>0}$, and let $c_r$ be the associated lacing parameter. Perfectness conjecture. The crystal…

Let $B^{2,s}$ be a crystal of type $D_n^{(1)}$, and let $B(k\varpi_2)$ denote the classical crystal of highest weight $k\varpi_2$. Let $D_{B^{2,s}}$ be an energy function on…

Lusztig's crystal-graph conjecture. The operation $*$ on irreducible components of the nilpotent varieties gives rise to Kashiwara's crystal graph.

Anderson–Mirković conjecture. For every MV polytope $P$ and every $j\in I$, $AM_j(P)$ is an MV polytope and

Let $\mathfrak{g}$ be of type $D_{n+1}^{(1)}$, and let $\widetilde{B}^{n,s}$ be the affine crystal constructed by the stated classical decomposition and affine operators. Spin-crys…

Let $L_\theta(\lambda)$ be the crystal lattice, let $c(L_\theta(\lambda))$ denote its bar-conjugate lattice, and let $V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}$ be the dual latti…

Let $B_\natural$ be the limiting crystal obtained from the crystals $B_l$, and let $g_l:B_l\to B_\natural$ be its set-theoretic embedding. For fixed $u\in B_\natural$ and…

Let $N \in {\mathbb Z}_{\ge 2}$. Fix positive integers $k_1,\ldots,k_N$ with $k_1>\cdots>k_N$, and elements $b_i \in B_{k_i}$ for $i=1,\ldots,N$ in the…

Admissibility decomposition conjecture. For any $\xi\in(P_{cl}^+)_l$, there exists a disjoint union decomposition, not necessarily unique,

Fix $n\geq 2$. Let $V(\Lambda_0)$ be the integrable highest-weight $U_q(\widehat{\mathfrak{sl}}_n)$-module, let $G^{up}(\mu)$ denote its upper global crystal basis, and identify…

Cyclicity conjecture. Given a configuration $m=(m^{(1)},\ldots,m^{(n)}),$ either every path in $P_h(m)$ is cyclic or no path in $P_h(m)$ is cyclic.

Dynamical-period conjecture. The dynamical period of $p$, namely the minimum positive integer $k$ such that $T_l^k(p)=p$, is equal to ${\mathcal P}_l$ generically, and is otherwise…

Level-restricted X=M conjecture. One has

Inhomogeneous X=M conjecture. There exists $b_0\in B_0$ with $\varphi(b_0)=k\Lambda_0$ such that

X=M conjecture. One has

Conjectural Kirillov–Reshetikhin module family. For every $(r,s)$, there exists an irreducible finite-dimensional $U'_q(X_n^{(1)})$-module $W^{(r)}_s$ such that:

Non-perfect crystal path conjecture. This tensor product is isomorphic to the crystal base of a certain tensor product module. This conjecture describes the representation-theoreti…

Particle-structure conjecture. The particle structure of the alternating vertex model should consist of the spin-$0$ and spin-$1/2$ particles created by $\psi^{(0)}(\xi)$ and…

Local–particle equivalence conjecture. There should be an isomorphism between the space in (SEMIPATH) and the space in (TENSORPRO).

Central renormalization conjecture. There exists an element $\Gamma$ in the center of the algebra such that $\widehat{T}_m^j$ has a smooth and well-defined behaviour in the limit…